I say you're over thinking. Either way you are well withing the limits of the device. I have never heard of or experienced an inverter failure in a genset.
The other problem you may create is that you could potentially overload the neutral in your cord if by going to the 120v setting the current in L1 and L2 is in phase. This gives you the possibility of having over 50 amps flow thru the neutral. (But this is very unlikely to occur unless you load everything on one leg on purpose.)
You would also have a greater voltage drop in your line cord if you were 120v only instead of 240v. As short as the cord is that's probably not a big deal but something to think about if you go running a long extension.
Leave it in 240v.