Well, if either Richard or Rich pops in they can give you lots of advice, but I'm going to bet that the first piece will be to try temporarily wiring
direct, with good sized wire, to a fully charged battery
. If it runs with new wire connected direct to the battery then the problem is voltage drop somewhere in your wiring
system (which seems the case since you only see 9-10 volts at the fan).
Only other option I can see is a locked up compressor, do you have an ammeter in the system? Can you see how much current
the fridge is drawing (even if other things are also on you should be able to tell by turning the fridge off then on and watching the difference in current