AMac

Hi AMac, the quality of power may be very dirty for the inverter to accept as you have mentioned already. Another well know trick is to change the menu option 3 to APL and not UPS. In APL mode the inverter is less fussy about the quality of the power. However, this all depends on how bad the quality really is in terms of waveform. 
I have only seen this inverter accept power from a 8kW heavy duty Diesel Gene. The smallies unfortunately seem to not have a good waveform. 

Since your generator is rated at only a quarter of the inverter's rated power, I don't think you want utility first output source priority. That will put the inverter into bypass mode, so the generator has to support all the loads. I would use SBU output source priority, so the inverter stays in battery mode, and the AC-in is used only to charge the battery.
I suspect that smaller inverters are more apt to have lower real-world power ratings than claimed as well. Plus the waveform issue.
I think if you are going to be serious about going fully off-grid, you'll need to pony up for a decent generator, perhaps one with an inverter built in, so it generates a good sine wave that the inverter can work with. And for the inevitable (even if rare) run of bad weather, the generator will have to supply most of the load, so it has to be big enough for that contingency.