Ted, you need to clarify how you are using your systems, but offhand, I'd say that your charger may be failing and your batteries are undercharged. Your refrigerator is running on DC and you need to leave your refer breaker and probably your selector switch on when you leave the boat. If the temp is just 60 when you come back, it's because your batteries haven't been charged while you are away or you are turning either the breaker and/or the selector switch off.
Your A.C./heat (I'm assuming reverse cycle) runs on AC and will obviously work just fine on shore power even when your batteries are low or dead. The DC power makes no difference.
Based on your Charles charger, I'm assuming you have a stand-alone inverter. (Edit: You call it an inverter and a charger on separate instances - so now it appears to me that it is a combo unit - Charles probably made crappy products since it seems they no longer sell or support marine products). We know nothing about your inverter, but we know that it doesn't charge the batteries. We don't know what you are running on the inverter or when (I assume you don't have AC systems, such as reverse cycle plugged into the inverter, but you must have some appliances that are used with the inverter. You should describe how and when you use the inverter because it makes no sense to use an inverter when you are plugged into shore power.
If you are on a mooring, there is no way that your batteries will stay charged with an inverter running AC systems or appliances of any kind without some charging source. Your Charles charger obviously does not function when you are not on shore power. There are just too may unknowns and you will need to fill in the blanks.
I'd say that you are not charging your batteries properly, or you are running systems without providing a charging source, or your charger has failed. In a year and a half, you may have kept your batteries chronically undercharged and the batteries have been killed during a short useful life. (Edit: I'd bet the charger is malfunctioning and needs to be replaced.)
Also, you never described what voltage meter you are telling us is 0. Is it a DC volt meter for the batteries, an AC voltmeter that will read the shore power voltage, or a volt meter on the inverter reading input on the DC side and/or output on the AC side. We need to know what you are looking at.