What does Jesus' being the χαρακτὴρ of the substance of God mean for Paul and his readership?

St. Paul is making what is an obvious (somewhat praphrastic) reference to, and development upon, the familiar personification of Wisdom in the Old Testament: namely, that 'theology' of Wisdom contained in the book of Wisdom of Solomon.

This book falls into the Wisdom Literature (Job, Ecclesiates etc.) of the Bible, considered canonical Scripture among most Christians (notably Catholics, Orthodox), but not among the Protestant and 'post-Reformation' communities.

This book, being in the Septuagint (Greek Old Testament), was in any case familiar reading to St. Paul, and is quite possibly the inspiration for his theology involving Christ being "the power and the wisdom of God" (1 Corinthians 1:24. cf. Wisdom 7:25-26). And I think, inarguably the source for His Hebrews 1:3 (and elsewhere) theology on the Word of God, Jesus Christ.

[cf. Wisdom 7:27: Hebrews 13:8; Revelation 21:5]

Firstly, the Greek of both, followed by how I would translate them:

Wisdom 7:25-26

25 ἀτμὶς γάρ ἐστιν τῆς τοῦ θεοῦ δυνάμεως καὶ ἀπόρροια τῆς τοῦ παντοκράτορος δόξης εἰλικρινής διὰ τοῦτο οὐδὲν μεμιαμμένον εἰς αὐτὴν παρεμπίπτει 26 ἀπαύγασμα γάρ ἐστιν φωτὸς ἀϊδίου καὶ ἔσοπτρον ἀκηλίδωτον τῆς τοῦ θεοῦ ἐνεργείας καὶ εἰκὼν τῆς ἀγαθότητος αὐτοῦ

25 For She is a breath of the power of God, and an emanation of the pure glory of the Almighty, on account of which no polluted thing can enter into Her: 26 she [Wisdom—v. 24] is the effulgence* of the eternal light; and a spotless mirror of the power of God and image of His goodness.

Hebrews 1:3

ὃς ὢν ἀπαύγασμα τῆς δόξης καὶ χαρακτὴρ τῆς ὑποστάσεως αὐτοῦ, φέρων τε τὰ πάντα τῷ ῥήματι τῆς δυνάμεως ‹δι’› αὐτοῦ, καθαρισμὸν τῶν ἁμαρτιῶν ποιησάμενος ἐκάθισεν ἐν δεξιᾷ τῆς Μεγαλωσύνης ἐν ὑψηλοῖς,

Who, being the effulgence* of His glory and the very aspect of His substance; upholding all things by the word of His power; having made purification of [all] sin, sat down at the right hand of the Majesty on high.

• ἀπαύγασμα—[the] shining forth [of something]; radiance; effulgence; refulgence (exactly synonymous with Latin refulgere). Only used here and in Wisdom 7 in the whole Bible! So it's almost impossible to be coincidence to find this word used in only these two places, speaking about the same Wisdom, in almost exactly the same terms.

A useful witness to its meaning for the early Church is the Latin Vulgate (4th century) translation, which interprets it:

Hebrews 1:3 Latin Vulgate

Qui cum sit splendor glóriæ, et figúra substántiæ ejus, portánsque ómnia verbo virtútis suæ, purgatiónem peccatórum fáciens, sedet ad déxteram majestátis in excélsis

Who, being the brightness of His glory, and the figure of His substance; and upholding all things by the word of His power; making purgation of sins, is seated at the right hand of the Majesty on high.

Between these three I think we find a common truth being related: Jesus, the Wisdom and the Power of God, derives directly, and intrinsically, His nature and being from the being of God the Father; He is 'begotten' from all eternity, just as light is 'begotten' from a fire.

He reflects—or displays Him—Him perfectly. That is, "the Son is the image of the invisible God" (Col 1:15; cf. Wisdom 7:26)—just as light is the visible aspect of fire.

The image of the invisible God tells us much. Because it is not speaking about a mere depiction of God. But rather and intrinsic element to the very substance of God, which is definitional to Him. Just as a body is defintional to a human (composed of invisible soul and body, although there is no 'compostion' in God).

"No man has seen God at any time: the Only-begotten Son, who is in the bosom of the Father, has made Him known" Jn 1:18).

"The big battle of the fourth century was whether Jesus was ομοουσιος (same substance) or ομοιουσιος (similar substance). What does της υποστασεως αυτου indicate about that question if anything?"

Well, της υποστασεως αυτου means "the substance of Him"—"His substance". So when you qualify or get specific with that, i.e. you say someone is the χαρακτὴρ caraktir (i.e. imprint, mark, or distinctive mark by which a thing is known; figure; likeness—image?) of someone's very substance, you are saying that the referent of 'caraktir' is too intrinsic to the referent of 'substance' to be anything other than, well, what Christians teach about the Word, or Son of God: an intrinsic natural relationship, a sharing of the one Being and Substance: God.

As seen from Wisdom 7:25, this Wisdom is the "emanation of the pure glory of the Almighty". He isn't a partaker, but rather someone who lives as the "emanation" or "effluence" and the "effulgence" or brightness—'visible aspect' so to speak—of the invisible God.

That is, the Word of God, Jesus Christ, is "in the bosom of the Father".

The word χαρακτὴρ in this context

It cannot mean simply, or merely, 'image-bearer of God' or 'copy of God', as you suppose, not only because of context, but because it is markedly (no pun intended) metaphorical/similaic in function: Jesus isn't actually an inscription of someone. St. Paul compares how conveyed-representation-of-person such as that on a coin, is kind of like how Jesus emanates from and 'shows forth' God, being "the image of the invisible God" (Col 1:15).

Jesus, or rather the Evangelists who records His sacred words, equate the words εἰκών (image) with what we would primarily consider a χαρακτὴρ (marking; impression; stamp; imprint) in Matthew 22:20 and Mark 12:16.

But then He further equates this χαρακτὴρ-bearing with representing the authority himself (Caesar).

In this context (Hebrews 1,and arguably Wisdom 7) the χαρακτὴρ of someone's very substance is more than the simple 'resemblance' or 'copy' of them.

Conclusion

The χαρακτὴρ of God's ὑποστάσεως is defintional to His eternal nature—His "glory," "power," His "upholding all things" etc. Thus, Jesus is ομοουσιος—"consubstantial with the Father".