Keld,
You have a lot of good advice here, but a little bit needs to be more specific.
Even if you bought the HR new off the lot, that does not mean that everything was done properly and all of you symptoms point to either a bad house bank or bad connections to a house bank.
Do you have a hand held - stand alone VOM?
If NO, go get one. It needs to be able to read to 0.1 volts at ~12.
Disconnect from shore power.
Put your head in the battery box, put the probes right on the battery posts (not the cable connections and there is a reason).
Now, have DW crank the APU (genset).
If it does crank, well, it is tough to diagnose what is working....
Does the battery voltage disappear? (Like go real low?)
=> The bank isn't charged or is defective. Try charging for a day and then try again. It can take a day to recover a really dead bank.
Does the bank voltage stay or drop just a little when the APU doesn't crank? (This is directly on the battery posts.)
=> Then you need to disassemble and clean all the connections to the batteries. And, then try the test again.
I used to work on boats before the depression and I saw this sort of thing all the time. Usually cleaning the battery terminals was a quick, but not cheap fix (if I was getting paid for it).
Matt