Dometic AC B59516.711 15,000 BTU, while on generator power the AC works during day while it hot but once the sunsets and it cools down, compressor kicks on it drags the gen down and shuts it off.

Something else besides the ac is stopping ( overloading) the genset
if it works well during the day, the compressor has to be working so the ac is not the problem
a 4000 watt gen set has a duty cycle load of around 3000 watts full time so any additional power requirement besides the ac will overload it
electric stoves , microwave ovens , hot water elements , anything that consumes power after dark will when combined with the ac will overload and shut down the gen-set