I picked up that you were joking about the shoulder mount. Obviously, you'll take the shot if you see him again. I'm not sold on the idea that this deer is doomed to a painful death though. He looks darn good to me and the wound looks fairly clean. Other deer will lick that wound and keep it clean and healing. I've seen them do it.
Two true stories to make you feel better:
Several years ago I was casting a scum frog over a farm pond weed bed and caught a huge bass. Big 'ol sow belly! I planned to release her but I wanted a photo. My camera was up at my truck so I ran a nylon stringer through her lip (away from the gills) and shoved the sharp end into the bank. Stupid! As soon as I stood up she swooshed out to the deep water and ripped the stringer out of the mud. I was sick about it because I knew she was going to die a slow death. Then over three months later I was fishing the same pond and caught a big but sluggish bass. As I lifted it into my canoe, my hand came up against the nylon stringer! She was a little thin but still strong enough to take a spinnerbait. I gently removed the stringer and released her. Never did get a photo.
A year before that, a buddy took a bad shot on a decent buck during early bow season, right before dark. He did not see the arrow hit but he heard it and tracked a sparse blood trail before losing it. He felt terrible. Then, in the heart of the rut he was rattling from a stand, nearly a mile from where he had wounded the buck. Suddenly a nice buck came charging down a hardwood hillside to investigate the rattling. My buddy said it looked like the deer had a flag above its head. It was not until the deer was in range that my friend realized that the flag was actually the veins of his old arrow, that had lodged behind the buck's jawbone over a month earlier! He made a belated second shot and killed the buck. On inspection, the buck was fully healthy, with only slight inflammation around the wound.