I've written a simple problem matcher, but unfortunately I haven't quite figured out where in the code you are generating the tasks.json, so I can't just do a PR for it.

      "problemMatcher": {
        "owner": "pde",
        "fileLocation": ["relative", "${workspaceFolder}"],
        "pattern": {
          "regexp": "^(.+\\.pde):(\\d+):(\\d+):(\\d+):(\\d+): (.*)()$",
          "file": 1,
          "line": 2,
          "column": 3,
          "endLine": 4,
          "endColumn": 5,
          "message": 6,
          "severity": 7,
        }
      }

The problem matcher works reasonably well, but it is not being triggered, when you use Processing: Run Processing Project, only when you use Terminal -> Run Task -> Run Sketch, so maybe some stuff for Processing: Run Processing Project would have to be changed.
Maybe the extension could work with VS Code's Run active file?