RE: Whats best to monitor battery voltage?
Personally, I recommend the Xantrex Battery Monitor, that is, if you really want to know the nitty gritty. Also known as the Xantrex XBM and if you really want to know what's happening, you can get the serial port option and connect it to your computer for some really in-depth monitoring. I know, I use it.
RE: Are these problems connected?
Your reefer (when using AC power, generally) works by a heating element that does the same thing your LP does when running on LP. It's a simple heater arrangement that takes the place of your pilot and burner when running on shore or genny power.
I suggest you plug in your reefer, check the switches to ensure they are set correctly for AC operation and then let it sit for about 10 minutes. Go to the access panel outside and feel around in the area where the burner lives and you should feel some heat being generated in that area.
If so, then you may have deeper problems but that's unlikely if it's working well on LP. If it's cold then you may have a problem with power getting to the element, or the element itself may have gone bad.
The element can be tested with a simple Volt Ohm Meter. If you don't have one, get one, it can save you much headache. Learn how to use it, they are not complicated.
If you can't test the element itself, try testing for AC being available at the socket your reefer plugs int, if it has one, by using a drop light or anything that can tell you there's power actually going there.
