battery full but no power?

No one is more in favor of solar than I am. I worked in that field my whole career, almost 40 years. But you cannot count on it being there any given day. That is the reality, you either need a couple of days or of batteryl storage or an alternative charging system. To run high reliability off grid smlolar systems in CO we would typicamlly have 4-6 sunless days battery storage. Our requirements aren't that critcal and can be met by just running the TV if needed and recharging from the alternator if you don't want to carry a generator around and don't need a/c.

There is no one best battery. Depends on how you use your pod, how you feel about maintenance,how you'll charge it and store it. Like all things in life there are tradeoffs. My recommendations are:

If you camp without hookups often and have the tow vehicle tongue weight capacity then 2 6V golf cart batteries are a good way to go. Flooded batteries you can put distilled water in are the most cost effective and tolerate charging well, but you have to check the water level every few weeks. Otherwise spend the money and get AGM batteries.Just don't overcharge them.

If you mostly camp with hookups and when camping without them don't use much power then get a single marine dual use battery, which will handle some cycling but probably have a longer lifetime for you because of chemistry differences. Flooded or AGM, same criteria as for the golf cart batteries.

If you never camp without hookups then just about any battery will do.

If you boondock and are a heavy user but can't tolerate the tongue weight then you either need to install a lithium system or move some weight rearward to counterbalance the heavy batteries on the tongue.

In all cases you will need to keep the batteries charged while in storage, but the golf cart batteries will need more frequent charging because they have a higher self discharge rate. Either disconnect the rpod from the batteries to get rid of the parasitic loads or leave them on solar or onboard charger all the time.

Take it one step at a time. First determine if your current battery is still good or needs to be replaced.

Then worry about how to keep it or a new one charged properly. You can do that via the onboard charger, by removing the battery and putting it in a charger at home, by using your solar kit, or by intermittently charging it and keeping it disconnected while in storage. Which is best depends on your storage and usage situation.

Car batteries can recover from being fully discharged once of twice because they are not called upon to provide much energy. All they have to dois produce some current for a few seconds.

It does sound like a either bad connection or battery at it's end of life. Either will create a high resistance which will cause the voltage to fly up and down as your solar controller attempts to regulate the amount of charging. Charge it and retest it or take we it to a shop and have them test it.