The problem is pretty simple actually. You have the exact setup Painter doesn't like. When you have a single Poly mesh with many brush strokes, Painter doesn't handle it pretty well and performances issues will occur.
I advise you to change your plane for a high poly one. This way, you'll keep the same UVs, but lose the poor performances. In your current project, go to
Edit > Project Configuration > Select the file > OK
I aslo advise you to uncheck the Auto-Unwrap and the Preserve strokes
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