Why not shoot one of the fawns?
The overall health of the herd would benefit by keeping mature does around who have already raised young.
The odds of 1 fawn surviving alone with mom is drastically higher than with having a sibling around.
I would be hesitant to shoot an unknown button buck but their odds of surviving their first year are low enough to put that aside.

So, shoot a fawn, easy dragging and more tender meat. Then you don't have to fret over killing a mother.