You would want 120V for typical loads.
Heating loads and larger motors can be 240V (of 120/240V split-phase) or 208 (of 120/208Y.)
If you buy three inverters then you can have the 3-phase, good for some shop tools.
I think you can generate 3-phase from two inverters which are 120 degrees apart, by using transformers. But a third inverter may cost less.
Or maybe running them with VFD is better. You get variable speed and soft start.
But cheap VFD, due to diode/capacitor front end, has bad power factor and creates harmonics. That upsets some of my inverters (AC coupled Sunny Boy) but hasn't been a problem for Sunny Island battery inverters.
You still have to convert to 120v for the other stuff in the house right? Do you mean it would be cheaper/efficient to just run 240v from the start and then transform it down to what's needed?
If you've got a transformer, this would let all watts of your 240V inverter supply single-phase 120V. Compare to two stacked inverters, each phase gets the power of just one inverter. Transformers do have a standby load. For instance, here are no-load voltage and current traces for a 9kVA toroid transformer. With 118Vrms applied it carried 0.67Arms, for 67W reactive power. That's mostly energy stored in the inductance (out of phase current), and how much is recovered by the inverter vs. dissipated somewhere would make a big difference in battery life. I'll have to check that some day, wanted to use this transformer to let slave inverters sleep, only operate master, to save power under low load.
Connect 240VAC to your 120VAC vacuum cleaner and see how much it likes it ?
It should really suck!
(universal motor spins faster with more voltage. Maybe not for long? Low-rider guys would put 72V into 12V hydraulic pumps.)
I think I did plug a floor sander into repair shop test outlet while it was set to 240V, and it smoked from the get-go. So there are limits.