I would begin by finding the cause of the loss of 12v. power. Without 12v, your furnace, thermostats, AC, refrigerator, and lights won't be working. You will need a handheld meter to correctly diagnose the problem.
Just from reading your symptoms, this is what might have happened: Shorepower is powering everything well and the generator is started. The automatic transfer switch senses generator power and switches to that source or a manual transfer switch is engaged. The power supplied by the generator is not sufficient for the loads or the transfer switch is failing. The generator shuts down and the auto transfer switch or inverter with transfer switch is called to supply power. The inverter can't draw enough power for the loads and either trips a circuit breaker or blows a fuse on the battery supply cable. At the same time, you may have tripped one or more circuit breakers on the AC load center or the converter.