Steve,
I had the same issue with my 06 PT the gen would not start unless the Cat was running. I put a volt meter across the battery cables at the gen, 13 volts hit the start on the gen and it dropped to around 7 volts. Problem was the cable from the front run box to the generator was too small wrong gauge and was routed much to long. Went to the local welding store and bought 12' of single O welding cable and copper ends. Replaced the cable and also the ground cable which also was too small, now it starts no problem. Don't know if that is your problem but they seem to undersize cabling most of the time.