They were very unlikely to put a loop vent in the wall. Instead the trap arm would have continued horizontal until a vertical vent.
See https://terrylove.com/forums/index.php?threads/p-trap-install-stubs-out-too-low.68736/ for how you would handle what you are suggesting. But I think what you are suggesting is a fair amount of work for little advantage.
Seriously, a wet-dry vac could catch water as you pull the trap. I think those plastic nuts will lift up, and you will be able to slide the U of the trap sideways. Also, you could probably suck that trap dry with a wet-dry vac on the sink drain from above. And if it can't, a smaller tube pushed down the drain into the U could be sucked on by that vac. Even a vigorously-operated plunger may be able to remove most of the water in that trap before pulling the trap.
What is under that cloth? Maybe there is already a clearance slot, for all we know. ;-)
Also, what is that strip maybe 5 inches from the wall? Could it be there is a removable panel to make access easy?
If you are in Houston or San Antonio, an AAV is probably not allowed. But otherwise it probably is. But why? What you have is going to be functional. Pulling that trap should not be a regular occurrence.