The a/c relay is NOT commanded on would be the reason swapping doesn't help--- so when you put the a/c relay in the horn relay slot-- does horn work? Sure it does, because you have a perfectly good a/c relay that is NOT being told to come on-- Jumping the socket turns the switch on that the car's computer says to keep it off- there is a reason for the "NO a/c" command. you need to find it-- most likely reason is low refrigerant charge due to a leak-- adding refrigerant would probably allow the system to command on the compressor. But you need to find anfix the leak first.
Also, there could be pressure sensor issues, or temperature sensor issues- this is where a scan tool would be needed..