What kind of golf cart battery has only a 108 amp hour rating? The common Trojan model T-105 has a 225 amp hour rating at the 20 hour rate and should give you 750 cycles of charge/discharge at the 50% depth of discharge recommended. Dennis is correct that the series stack of two 6 volt batteries result in the same amp-hour capacity at twice the voltage...supplying the needed 12 volts to your compressor modification in your refrigerator. But 225 amp hours would give you more run time than you are experiencing for sure.
Do not put 12 volt lead acid battery in parallel with your existing golf cart battery stack -- mixing battery types, manufacturers, ages and conditions in a parallel charging setup is a very bad idea. Even if you had room and decided to use a second pair of golf cart batteries in parallel with the ones you have now, it would be best to replace all four to ensure the match of manufacture date and chemistry/condition is optimum.
Lets look at the voltage drop you are experiencing. If you have 10 gauge wire and the length is 35 feet, the resistance of that wire is only about 0.035 ohms (this wire is rated at a resistance of only one ohm per 1000 feet of length...) taking the wire resistance of 0.035 times your peak current draw of 9 amps yields a voltage drop of just .315 volts. Because this is under half of what you observe as voltage drops, your connections must be adding in substantial resistance of their own. If you have used crimp on terminals, did you use quality gas tight terminal parts and a quality ratchet crimp tool specific to the manufacturer's specification?
What is the terminal voltage of the stack at your batteries when at rest and when the compressor is running? Perhaps you have some oxidation in the battery terminals themselves, which would cause added resistance and give poor performance. Lead oxide is colorless and hard to remove with just a wire brush tool, it is best to scrape the terminal posts and the mating connection to ensure a low resistance connection. You can make the voltage measurements at the battery posts and again at the cable terminals to see if there is any difference under load - any noticeable difference would be a sure indication of oxide on the terminals.
Let us know what you find...
2007 Tour Master T40C
Acura MDX toad
"It takes a great deal of time to recover from any improvement..."