I'm not sure whats going on there, but I would take them all apart and connect only 2 batteries for 12v. Charge the 2 batteries, connect inverter and see what happens. If it goes down fast connect the next 2 batteries and test those same way. No need to have them all connected at once, just what ever makes the volts your inverter needs.
I can't really tell in that picture, but do you have 2 cables comming off those batteries for both pos and Neg connection each?? Like charging from the left side cables and inverter on the right side cables? Perhaps 1 bad connection on one wire could be the problem too then.
What I am thinking is maybe you are charging the 2 sets to the left, bad cable connecting those to the set on the right would mean your getting a charge on 4 batteries but trying to pull power off only 2 dead ones. Is this case you may have batteries barely raising to 12V on their own sitting Idle but any load at all kills them. This would apply to the middle set also if the bad connection is between the first set. Have you checked all 6 and they all have only 5.89V?
Have you tried running the inverter while charging? If your putting out 60amps from that alternator into those batteries then you should be able to run the inverter off that while charging if all connections are correct. If the inverter does not run a small load while the alternator is running then something is not right like bad wiring?
60amp from the alternator while charging should run nearly a 720 watt load even if the batteries are very weak, I have done this at a remote house I was renting for awhile. Batteries to low for inverter, charge off running car with jumper cables run inverter same time just fine.
That was a 5K inverter with small load of CFL lights and a full size frigerator and radio. Not sure the amps on the car altenator.
I don't know anything about your batteries but I find it hard to beleave all 6 would be shot and holding about the same volts. I would first test if the inverter runs a load while the alternator is running (charging) and then test only 2 batteries at a time charging them and loading them.
Good luck