According to the specs the CCA for the Firefly is 720 Amp. So the BT should be fine having 5*720=3600 startup amps available.
I don’t buy this generator story as your 100 amp battery charger cannot respond quick enough for the solenoid to make a difference. And besides it can only supply 100 amp where the startup current for the BT is probably closer to 1000 amp.
When running the BT takes about 450 Amp,, about 80 amp per battery. If your batteries cannot supply that, you need to replace them anyway...