[raymo]

Hi again Mike. You can charge a battery or a bank of batteries using several
charging methods simultaneously. Solar panels do not allow reverse current,
and battery chargers are protected against reverse current. For example, I
can have my car engine running [ so obviously the alternator is charging the battery]whilst the solar panel is still connected to the battery.
I had forgotten how much cheaper open frame tradesman's style generators
are than the enclosed quieter recreational style ones. I don't know how many amps you are looking at contributing via the generator. Recreational ones all have a 12v charging outlet of 8amps or a little more obviating the need for a separate charger. Any recreational generator will charge at 8 amps, doesn't have to be a heavy, large noisy 3kva unit. My 750w charges 8amps using a minimal amount of fuel.
I note the model in your link doesn't seem to have this feature.
raymo