While it does not cause your delay problem, both fans on your car should always run together, at the same approximate speed. Never one by itself.
Never.
You have a fan that is failed.
Running your AC with only one fan will result in higher than normal pressures and temperatures that your compressor has to deal with.
In every case I am aware of but one, the fix was to replace the AC compressor. One owner (in Texas, I think) still had the problem after replacing the compressor.
If you do replace your compressor, please consider shipping it to me. I would like to do a post mortem on one of these.
One hypothesis is this. The compressor's displacement is varied by a wobble plate. The wobble plate is controlled by the difference in discharge and suction pressure of the compressor, ported by an electric Refrigerant Control Valve. It depends upon compressor discharge pressure being higher than suction pressure.
From the reports I have been given, the RCV was working, but the compressor was not responding.
Conjecture:
The hot cabin is making the highest pressure in the system be inside the cabin, prior to engine startup. It is not unusual to get 130 or more degrees F inside the cabin. It is seldom that hot under the hood before the engine is warmed up. This gives you a reversal of pressures. The suction pressure is actually higher than the compressor discharge pressure. So when the RCV opens, instead of putting high pressure from the compressor discharge to the wobble plate piston, it puts low pressure to it. This puts the compressor hard on the minimum displacement stop, holding the compressor at minimum output. It stays this way until the cabin cools off some, and the condenser coil/compressor heats up some, creating the normal condition of compressor discharge pressure higher than suction pressure. Now the pistion will move the compressor to full displacement, full output. This condition will remain during operation and shutdown until the cabin once again heats up more than the condenser/compressor - which can take hours as the radiator is pretty hot.
If the minimum displacement position was at a bit higher condition than it presently is, perhaps there would be no delay. Shimming this stop may or may not cause a problem during winter operation with these compressors. But I bet it would be ok in the South.