I just wanted to report that following replacing the LP tank and regulator, which is admittedly a $350 fix, I have found it much, MUCH easier to get the fridge to light on propane.
My tank and regulator were having issues, so it may have affected this. But whereas lighting the fridge, even after running it on 110v, was always a hassle, it now lights up in just a few seconds with a quick gas prime, a few pumps and one or two pops of the ignition.
It still is stubborn if completely cold, but I have managed to light it relatively easy after running it on 12-volt for a few hours and then going to gas.
I don't know if it was rust in the tank, moisture in the tank, plain age or what, but the new tank has now made the fridge light the way you would have expected it to in the first place.