assuming you have a Surveyor TT, since you don't post which model you have, you CAN'T run the fridge on 12vDC.
your fridge runs on either propane or shore power. the propane side will use 12v battery power to light it and to power the electronics. but it doesn't use 12v to cool it. you would need a 3-way fridge for that and they are only in popups and truck campers and other very small trailers.
the 110vAC side will only work when you have hookups.
as to the thermostat, simply Google "Hunter thermostat install".
this is a common mod that tons of RV owners do. simply by a battery powered digital thermostat and install it. the Hunter and Honeywell ones are the most popular.
and Zoe the Wonder Dog(R.I.P.)
'07 Roo 23SS pushing an '07 Chevy Avalanche
Equalizer WDH and Prodigy BC