can you turn on the entertainment system in portions, IE: tv first, then receiver, then cable box?? all these electronics have caps in them that may have large spikes of current upon initial start up.
also, have you closely monitored battery voltage during start up? if it's dipping hard from the surge, it may be shutting down due to the low battery shutoff built into the inverter.
have you checked all the connections as well? pull the wires out, clean then, put a bit of antiox on then, reinsert and toque down snug.
either way, that inverter should probably have thicker pure copper wire.
with 4' total needed, it should be pretty cheap. i've got plenty of scraps of #3 of that length, but they're at my ranch. otherwise, i'd just send them to ya.
adam