The ATS depends on generator voltage to determine whether to engage or not. If it falls below a certain threshold voltage, the ATS decides the genset is not active and it disconnects. It's conceivable the ATS is making a mistake, but far more likely that the generator voltage has actually fallen below the critical value. My bet is that the compressor start-up surge is overloading the genset momentarily and it ceases to put out power, so the ATS disengages. Then the genset rests and begins producing power, causing the ATS to resume generator mode.
The a/c compressor does what is known as a "hard start", consuming a huge amount of power for a second or so to get going. A compressor in poor working condition will be even harder to get going. Some RV a/c units have a "hard start capacitor" to give it a power boost for starting, and these are available as an add-on for those that do not. Your report that it has become progressively harder to get the ac/ started suggests that the capacitor on yours is dying or the compressor itself is getting weak. My first step would be to replace or add the capacitor on yours.