House 240 backup power to junction box from 120 inverter?
Note: this answer is purely technical, legal issues are something you would have to check with whatever code your jurisdiction enforces.

From a technical perspective there are a few things to be consider.


Currents will sum rather than cancel in the neutrals of "multiwire" circuits. This is unlikely to be an issue with such a small inverter but it's something to be aware of if you try to do this with something bigger. 
How will the inverter respond to overloading if you forget to turn off high power equipment? presumablly it will trip out but tripping out frequently is unlikely to be good for the inverter.
If you have any equipment that uses both hots and the neutral how will it respond to this situation. 
Whatever you do make sure you don't backfeed the grid. There should be a suitable switching/interlocking system in place to make sure that the grid and inverter are never connected to each other.