Is it Possible? Making texture UVs bulge around texture mask.

How could you could make UV coordinates bulge outwards (radialy, away form painted areas of mask)? Where black would be the largest bulge and white would be no bulge. White value = 1X uv coordinate size black value = 2x uv coordinate size.

if im reading this correctly your not looking to bulge anything, the uv map should not change, what you want to do is tessellate your mesh using the material and then use a displacement map to drive your bulges,

Im looking for a way to “pinch” some areas of a 2d texture, and splay out other areas (in engine).

Would flow maps without any speed/movement work?

yes, Flowmaps would work, albeit crudely. There are some bad points about them: they are very difficult to control precisely, And show stair-stepping artifacts when seen up-close/at low resolution.