First, an appropriate battery bank and inverter can run large AC loads no problem. We run everything off our inverter/batteries - A/C, dive tank compressor, water heater, 120V 35gph watermaker, etc. We don't have a genset and are rarely on shorepower.
Assume for now the OP's inverter and battery capacity is not the problem. That might be step 2, but not the issue right now. He states he gets no voltage at all on bus B when he parallels the inverter on bus A, but this works for shore power and genset. He also states no breakers or fuse are tripping.
Without a circuit diagram, it is difficult to say what the issue is. It sounds like it should work, but the devil is in the details. Where are the lockouts for the shore power and genset located? Where is the parallel switch in the panel wiring? How is the inverter wired to bus A, and is the shore/genset wired through the inverter's transfer switch?
These circuits are commonly wired with a lockout selector for shore/genset power. The lockout switch output then usually splits with one leg directly to Bus A to power high load items only from shore/genset, and the other leg to the inverter transfer switch. The inverter is wired to Bus B for low load items, with a lockout switch to lock out the inverter if the two buses are paralleled. So normally, it is not possible to parallel the inverter on Bus B to Bus A, and the inverter is not normally connected to Bus A. It's possible the OP's system has a 3-way selector switch on the AC input - shore/genset/inverter, with all inputs going to Bus A. If so, then this should work as expected.
Mark