Interlock and backfeed setup with NEMA L6-20
Running the house off the L6-20 receptacle (without the neutral) would be almost sure to start wrecking things on your 120V circuits in short order, blowing light bulbs, refrigerator, etc.  And at the worst possible time, during some emergency.  

Since there are 120V receps you know there's a 120V tap in there that can be used to supply line-to-neutral loads.  However there's a chance it is too small to carry the full neutral load for this application.  I am not sure, but from that wiring diagram but it looks like the neutral tab on the 120V receptacle is the neutral-ground bond.  

I think it would be possible to modify the wiring and install a L14-30R receptacle, but it wouldn't be a modification I'd be comfortable making.  The alternative would be to use a transformer between the generator and the backfed breaker, but at that point maybe you just buy a different generator better suited for the application .