We have used our inverter to power our fridge while traveling down the road. Running a generator to power a small fridge is overkill in my opinion, and if it is not an onboard genny it would be too much hassle to try and rig a portable unit to do the job.
Our fridge required about 325 watts from the inverter to cool, there would be some drain on the battery as the small diameter wiring from truck to trailer was incapable of delivering enough charge to replenish what the fridge was consuming.
Not an issue if we were going to a site that had hookups, but for dry camping the battery would be down a bit upon arrival. I don't think your small fridge requires that type of wattage to run, check mfg. specs in order to get an inverter capable of doing the job.
The preferred method of installing an inverter is near the battery due to the voltage drop incurred with DC, then routing the wire to deliver AC to the appliance in question. It is doubtful the DC wiring near your fridge will be capable of delivering the power required by the inverter.
We have installed a couple of inverters since purchasing our rig new 8 years ago and they have served their purposes well. We have ceased using our inverters to power our fridge while underway, they are primarily used to run our HDTVs, HD satellite, DVD/game players, and other miscellaneous items when camping off the power grid.
