I can't say exactly what I would do.
On one hand I know that a lot of severely wounded deer survive for years and years (three legged deer, deer with completely busted limbs, huge wounds) as we often take them long after they have healed.
On the other hand there is obviously the question of what is humane? Letting something that will most likely die suffer just because I don't have another tag?
But on the third hand

the coyotes and the ravens have to eat too. Thousands (maybe millions?) of deer die of "natural" causes every year. The only thing that makes this one special is that I was there to see it.
I'm leaning towards probably shooting the wounded deer (without a valid tag for it that is) assuming that I know it's an easy shot, it's a doe or a fawn and if I think I have a chance of making my case with any EPOs I might meet.