Powering both legs of 120/240V with a 2-pole backfed breaker that has both poles connected to a 120V source can work, but I don't think it should be done with a source (e.g. inverter) that can deliver more than 20A. Routing it through a 20A fuse or breaker before it branches to the two poles of a backfed breaker would be good.
If not limited to 20A total, it could deliver 20A to each leg of the breaker panel and might put 40A through a single neutral wire (depending on how house was wired), which is a fire hazard.
If you do backfeed main panel from inverter, turn off the breaker which feeds inverter from main panel (assuming there is one) so inverter doesn't feed itself.