In your case, I would use a Victron Autotransformer. You can wire them to step up 120V input to 240V split phase.
If wired inline after the load center and before the disconnect box, it will be used regardless of the power source. It also will have its own secondary breaker. No need to bother with a separate inverter, as long as your primary inverter has sufficient capacity.
2000 Country Coach Magna 40',
4380W solar, 22.8kWh LiFePO4@48V, 450AH AGM@12V
2020 Jeep Wrangler Unlimited Rubicon 2.0T, cloaked on 37x13.5s