Reading the manual you need 11.7V of more for the switch on voltage, you have a reading of 11.8V so this should be sufficent all being it very close to the cut off.
Its going to look to pull around 15A at 12V so good chance is the battery voltage will drop bellow 11.7V on startup.
I was initally wondering if the inverter was trying to take DC and pass it through as AC, but it looks like it runs on DC, it also appears its not needing to do anything other than pass through the 12V so there isnt a transformer at fault.
I would say the issue is the battery, if you can pick up another 12V or borrow one from another boat plug that in directly, check the voltage to see if that is above 11.8V and if the fridge starts..
Secondly if that still doesnt start put the two batteries in series and plug it into the fridge and see if that allows the fridge to power up on 24V.
It could be simply your volt meter being 0.2V out, or the tollerance of the fridge not being 11.7V but 11.8V for it to start, and as the battery has aged the voltage has dropped.
The other thing you may notice is that once the fridge cools on mains, the fridge will start working again on the original battery as its not needing to pull as much current on startup.
Let me know how you get on trying with another battery (borrowed or otherwise)