Ever hear of Zyrtec, Benedryl, Claritine?? My daughter has allergies and she takes Zyrtec whenever she comes to visit. Works great for her. It doesn't sound like this family has exhausted all avenues. There probably won't be much sympathy from BPO if that is their only reason for re-homing him.
How long have they had him? They just now discovered allergies? Why the 'next-week' deadline? They've had him this long. Can't they keep him until they find him a decent home however long that takes?
This is not directed at you. I know you are just trying to help, but goodness, look at that face!
One can only pray for a good outcome in this situation.