Could one choose/decide not to have (or to give up their) free will?

OP: "he does not use the term "free will", but it seems not hard to interpret "free" and "freedom" in a adequate, compatible manner."

A note from Schelling's Philosophical Investigations into the Essence of Human Freedom ties things together. It refers to “the system of the equilibrium of free will”, e.g.

the common concept of freedom, according to which freedom is posited as a wholly undetermined capacity to will one or the other of two contradictory opposites, without determining reasons but simply because it is willed, has in fact the original undecidedness of human being as idea in its favor; however, when applied to individual actions, it leads to the greatest inconsistencies.

… for example, anyone has the power now to draw back or extend his arm without further reason; for, if one says, he stretches his arm just in order to prove his free will, then he could say this just as well of when he draws it back; interest in proving the statement can only determine him to do one of the two; here the equilibrium [Gleichgewicht] is palpable (page 48)

And so to the note on page 162:

Schelling applies the notion of free will as “the system of the equilibrium of free will” to intelligible being, expanding the analysis begun with the discussion of Buridan, and it might be useful here to refer to Heidegger’s interpretation of this initial notion of indifference, as the ultimate form of freedom if freedom is defined solely as a kind of “freedom from,” in connection with his generally incisive taxonomy of the varieties of freedom discussed in the Philosophical Investigations. Heidegger notes:

If freedom means man’s complete indeterminacy, neither for good nor for evil, then freedom is conceived merely negatively, as mere indecisiveness, behind which and before which stands nothing. This in-decisiveness thus remains nugatory, a freedom which is anything else but a ground of determination; it is complete indeterminacy which can never get beyond itself. This concept of freedom is again a negative one, only in another respect, familiar in the history of thought as the libertas indifferentiae, the seventh concept of freedom in our count. (Heidegger, Schelling’s Treatise, 102)

The other six are:

1. Freedom as capability of self-beginning [Selbstanfangenkönnen].

2. Freedom as not being bound to anything, freedom from (negative freedom).

3. Freedom as binding oneself to, libertas determinationis, freedom for (positive freedom).

4. Freedom as control over the senses (inappropriate freedom).

5. Freedom as self-determination in terms of one’s own essential law (appropriate freedom), formal concept of freedom. This includes all of the previous determinations. (Heidegger, Schelling’s Treatise, 88)

And of course

6. that “freedom is the capability of good and evil.” (Heidegger, Schelling’s Treatise, 97)

In Heidegger's On the Essence of Ground, in which he indeed mentions Schelling, he takes freedom on to an altogether more primordial grounding.

Kant apparently showed little interest in the "principle of reason," even though he explicitly discusses it both at the beginning⁸ and toward the end⁹ of his philosophizing. And yet it stands at the center of his Critique of Pure Reason.¹⁰ Of no lesser significance for the problem are Schelling's Philosophical Investigations concerning the Essence of Human Freedom and Related Matters (1809).¹¹ (page 99)

The ground that springs forth in transcending folds back upon freedom itself, and freedom as origin itself becomes "ground." Freedom is the ground of ground. (page 134)

And in On the Essence of Truth, page 336, he makes it clearer how things stands with freedom.

if ex-sistent Da-sein, understood as the letting-be of what-is, sets man free for his "freedom" which confronts him, then and only then, with a choice between actual possibilities and which imposes actual necessities upon him, then freedom is not governed by human inclination. Man does not "possess" freedom as a property, it is the contrary that is true: freedom, or ex-sistent, revelatory Da-sein possesses man and moreover in so original a manner that it alone confers upon him that relationship with what-is-in-totality which is the basis and distinctive characteristic of his history.

So this is the type of freedom Sartre means when he says we are "thrown into freedom", as we find ourselves in being-in-the-world. Furthermore …

The essence of truth has revealed itself as freedom.

This is the ex-sistent, revelatory "letting-be" of what-is. Every overt mode of behaviour vibrates (schwingt) with this "letting-be" and relates itself to this or that actuality. In the sense that freedom means participation in the revealment of what-is-in-totality, freedom has attuned (abgestimmt) all behaviour to this from the start. But this attunement (Gestimmtheit) or "mood" (Stimmung) can never be understood as "experience" and "feeling" because, were it so understood, it would at once be deprived of its being (Wesen) and would only be interpreted in terms of, say, "life" and "soul"—which only appear to exist in their own right (Wesensrecht) so long as they contain any distortion and misinterpretation of that attunement. (ibid., pages 338–339)

"If to be free meant to be its own foundation, it would be necessary that freedom should decide the existence of its being."

This is the crucial element in your question. This is the presumption that leads to the apparent contradiction of using free will to decide not to be free. We do not create free will, we experience it. To have free will does not encompass the choice to have free will.

Imo, viewing free will as a burden is a punishing way to live. :(

Could one choose/decide not to have (or to give up their) free will?

If we suppose that one in fact does have free will, then one can give it up by suicide. Or, somewhat less permanently, by drugging oneself comatose. Conceivably, one might be able to will oneself into a psychosis so profound that it effectively negates their free will.

But I take Sartre's point to be that freedom (free will) is a fundamental characteristic of what we are -- living, conscious, sane human beings -- not about what we do or can do, short of changing our own nature in some fundamental way.

There's also the related sub-question of whether a "No" answer (either in the manner of Sartre, or in another one) would mean that (naïve notions of¹) "free will" is paradoxical/incoherent/self-refuting

No, it is not. Certainly not in any sense that I would construe as naïve. Sartre's "freedom", or what most people would mean by "free will", is not absolute freedom to accomplish anything and everything conceivable. It is not inconsistent that there are things that are beyond the extent of free will. That freedom is not absolute is one of the points Sartre is making in the text you referenced.

My ten cents..

Sartre's thesis in "Being and Nothingness" is based on a phenomenological study of consciousness that distinguishes between modes of Being: a) conscious beings-for-themselves and b) inanimate beings-in-themselves lacking consciousness.

The phenomenological study of consciousness makes evident the fact that consciousness is continuously and dynamically directed towards something else. In other words, it has no fixed essence (ie it is nothingness), it is free to continuously define and redefine itself.

That makes beings-for-themselves free agents (having a form of libertarian free will), which are not defined by fixed essences (there can be constraints imposed by the facticity of the moment nevertheless), unlike beings-in-themselves which are inanimate things lacking consciousness, and thus freedom.

Therefore, since beings-for-themselves cannot become beings-in-themselves, a person cannot (consciously) give up one's free will, one's freedom. We are then, as Sartre says, "condemned to be free".

Let's stick with simple logic for a moment. If one could choose to give up one's free will, that would mean — from that point after — that one could not choose to take it back. Such a one would not have the free will necessary to choose to regain free will; they would have become (with that initial choice) an automaton with no capacity to do anything beyond what is instructed.

This isn't what we see in the world. People frequently choose to give up freedoms, as in going to a job every day and doing what's expected of them; sometimes they have their freedoms taken away by others, being turned into slaves or held as prisoners. But they do these things for cause, and can always choose to resist, reject, or oppose the restrictions. Trust me, many, many, many unsavory people would love to find a way to get others to give up their free will, so they could have a sufficient supply of non-problematical workers, servants, sexual partners, or what they will for their profit or amusement. So far no one has managed to accomplish it, but not for lack of trying.

Philosophy is a type of kit that you can build whatever you want with it. "Researching mind while not event leaving sofa", or "basic types of thinking, like color and sound are linked to each other under the hood". So, we can build architectures and call it philosophy of goverment, make up our own minds, and draw imaginary animals and discover that it is a type of technology, like Rafael's "pigeons in Saint Peterburg at winter"

Cogito, free will is an integral part of personal identity, defines who you are, via your choices. If you give up your free will, you would lose your identity. In other words, it maybe possible to surrender your freedom, but you may not desire to do so.

Yes it is possible. It is known as surrendering to the will of God. It happens gradually when you get to know Self and realise there is lots of weird things going on in me. All the innate skills, the intuition all the stuff although originates in me but the source isn't me. There is an entity within that's not me and ACTUALLY THAT ENTITY IS THE SOURCE OF EVERYTHING IN ME. My creativity, my intelligence, MY WHOLE being is nothing but a reflection of whatever happens in that Entity. So the illusion of control shatters via Knowledge of Self, THE GUY IN CONTROL, IN ME, ISN'T ME. THE WILL TO WILL BREAKS...