Will shore power keep bateries charged?
While it is true that converter technology has improved in recent years, Gruffy is the only person that I know of who says that they are good at charging. I read tech. columns in most of the RV magazines and also have several professional RV techs among my friends & everyone that I know with a background in RVs feels that they are "so, so" chargers at most. I have been in electrical work for 40+ years and an RVer since 1972.
An RV converter is designed to do a different function than does a battery charger. It's design is to be able to supply 12V-DC power at varying load levels for a constant period for a long time. And the design is to do that without over charging, or cooking the batteries that are connected to the system. The reason that your tow truck doesn't over charge is that the battery is on a seperate line from the other DC loads and it gets a different voltage, depending upon it's state of charge. A wet cell battery actually varies from about 10.5V discharged to 13.6V charged. To get the battery to the charged state, the charger must increase the voltage to about 14.5V. While there are converters on the market that will do that and that do have a seperate charging circuit with a variable voltage, very few come in an RV as standard equipment. Most converters come set to a continous 13.5V or there about. That means that the battery will be charged to nearly full, but not quite, and it will not reduce the voltage once the battery is full, as your tow vehicle does.
If you have an inverter, they are typically installed to replace the converter and they can both draw 12V-DC from the battery to make 120V-AC or take the 120V-AC from shore power and make 12V-DC. They are much more expensive than a standard converter and most of them do have a good battery charger built into them. Many of them have a better battery charging circuit than do lower priced battery chargers. So the answer depends upon what your trailer is equipped with.
The other part of the question is, how long will it be stored and in what kind of weather? If it only sits for short periods and the weather is not really cold, I would probably not keep it plugged in unless I was also keeping either the refrigerator on, or a small electric heater in the RV. On the other hand, if you live where the RV is winterized and not used for several months, I would then use an external battery charger to be sure that they are fully charged and then lift the negitive cable to isolate them and not plug in the shore power. But if you do have an inverter, check the specs for it as some of those do have built in battery minders that will keep your battery up, by changing voltages appropiately. A converter might have that, but probably does not. In extream cold, it might pay to remove the batteries if they are not used for several months, but battery designs today seldom justify that much work. If your storage periods are pretty short and you use the RV all year so that it only sits for a month or less, then it probably is a case of doing what ever is most handy for you. But no matter what you choose to do, check the electrolite levels frequently to make sure that they don't boil away. When they are low, add distilled water to bring the levels up. Do not use demineralized water as it isn't the same thing. Once you have done this for a time you will soon know how often the levels need to be checked. But checking levels at least monthly when plugged in is important to battery life.