The refrigerator in question is a Dometic model DM2652 (two way refrigerator, made in 2010.)
I tend to always leave my fridge running on gas, regardless if I'm at hookups, running a generator, or just boondocking it. Even though I have screens on it, it helps ensure that the burner doesn't get plugged by a small spider or whatnot.
Is there any reason to run it on electric, other than to check that the electric side works? To my knowledge, since there are no moving/burning parts that bugs can crawl into with the electric burner, there isn't any real need to fire that up, but I could be mistaken.
I'm not really worried about propane use, and from my experience, the refer uses very little anyway.
For me, propane doesn't really matter. Since there are no nearby filling places, I just switch out the bottles before I hit the TT in the winter, and every month or so in the summer for the barbecue grille. Here in Texas, I've had some places charge $7.50 for a 20# tank exchange (with a coupon), so it isn't very pricy to swap the tanks out.