Beautiful pattern Nathan!
The paper towels is the "fix" I use exactly for the glue squeeze out you mention. It will keep the squeeze limited to filling the hole it came through. Small enough that I never have had issues with finish, even with porous burls.
Best fix is to get just the right amount of adhesives for sure or have the veneer surface covered with a non-permeable material like I use with the shelf contact paper. Of course this forces the glue out to the sides, but if the cauls are flat, the surface will also be flat though the glue layer might be a bit thicker than optimum if it can't fully expel the excess in the clamping process.