The signal to the compressor clutch comes from the computer. Just turning on the AC switch does not automatically turn on the compressor. There are a bunch of signals the computer uses to determine when the compressor should run.
I don’t have the wiring diagram for your particular car, I do for my 97 Accord. Unfortunately I cannot find the diagram for the AC section of the computer other than the the wire to the AC compressor clutch comes from the computer. I do remember seeing it for out 93 Civic and it had about 10 signals that it processed before allowing the compressor to come on. I don’t remember what all of them were and don’t remember if engine temp was one of them, but I suspect that it was.
Since running the AC causes the heat load on an engine to go up, your Acura may be shutting off the compressor when it detects a higher engine coolant temperature in order to protect itself.
BTW, this next piece of info is not really recommended anymore for safety reasons, so be careful. When the engine is cool enough to rest your bare hand on the radiator cap, remove it. Then run the engine until it gets hot. You should be able to see some circulation of coolant in the upper tank at idle. I don’t know it this will help you because you would have had to see this when the pump was known to be working good to compare with now.
I have to say that I would be a little disappointed if the pump is bad. I have had waterpumps go 2-300k miles for me with out a hitch. But since you need the timing belt, definitely do the waterpump, it is recommended. You have been changing the coolant every 4-5 years haven’t you? You have not ever used a chemical flush in your coolant system have you?