One of the first mods we did on our new rig was to install a large inverter, tried what you are asking about a couple of times and found that in our case we did not get enough charge from the alternator via the 7 pin cord to keep up with the roughly 325 watts the fridge required when in AC mode.
We noticed TT batteries were taking a hit while underway arriving to our destination with less than fully charged batteries. No biggie as we normally just set fridge to auto mode which of course switches to propane when traveling down the road. Our fridge cools better on propane anyway.
