[KISS]

To re-iterate, you have a 240/120 V 4-wire circuit and a 120 V circuit.

The only POSSIBLE thing you could do is to convert the 120V inverter output into a 240 center-tapped output by means of a transformer.

Two high rated 7.5 KVA instead of 2, but this is similar to what you would need with correct outdoor enclosure, etc. SQUARE D 7S6FIS Transformer 7.5kva

Grainger is Wholesale only.

[sailfast]

That's more trouble than it's worth. Thanks for your advice. Guess I'll stick with the status quo.

[KISS]

True, but you might be able to pick up the required transformer surplus - cheap.

You need 2 KVA with a 120/240 input and 120/240 output such that you get 120 V in and 240 CT out.

[Stratmando]

If not mentioned, Jumping both hots could cause overloading on a Multi circuit(2 hots sharing 1 neutral).
If all you need is 120 for power outages, perhaps 1 leg would do, the one with most emergency circuits. Swap necessary ones from other leg to that phase as well.
When placing a circuit on the opposite phase, be sure to place other multicircuit wire on other phase.

[sailfast]

If not mentioned, Jumping both hots could cause overloading on a Multi circuit(2 hots sharing 1 neutral).
If all you need is 120 for power outages, perhaps 1 leg would do, the one with most emergency circuits. swap necessary ones from other leg to that phase as well.
When placing a circuit on the opposite phase, be sure to place other multicircuit wire on other phase.

I don't see how this could cause circuit overloading. As I described earlier, the Yamaha generator is only capable of 2000w peak and 1600w sustained. Unless there's something I don't understand, the worst-case is I overload the generator and pop the protection circuit.

[Stratmando]

Then you're fine.