I’ve been thinking it may be an issue with the reed valve

No such thing as an inexpensive 12v cooling system that works well. Your best bet is to fix the generator (or replace it!) and run that thing while driving, to power the Roof A/C. (I do this regularly with an inverter-style gen on my rig’s back cargo rack, and in tandem with the cab A/C, it can keep the whole rig comfortable on 90+ days.)

No, it's  not built in. The generator is located in an outside compartment and requires access  from the outside to start and stop it. Plus we've never used it since it's  in pretty bad condition.