Scamp does not offer dual batteries or
solar, even as options, so those at least are owner modifications. Which leads me to wonder what other modifications have been made... Is there an inverter? What year is this Scamp?
Do we actually know the batteries are fully charged after plugging in to shore power? Have you checked the voltage? What type of batteries are they? Conventional, AGM, lithium...? 6V or 12V? If conventional, pop the caps and check the electrolyte level. Look full? Connections clean and tight?
As said, there really isn't much in a stock Scamp that would completely drain a dual
battery system in good condition within a few hours, but in this case there seem to have been changes. Unless you know your way around RV 12V systems and are conversant with the use of a multi-tester, you may want to consider taking it to a professional.
I'm not so concerned about your tow vehicle not fully charging the batteries. Very few can. Limitations include alternator output, lots of electrical demands in newer vehicles, and small gauge wiring for the trailer charge line. One warning, though: towing with the
fridge on 12V will typically end with warm food and a dead
battery.
Solar is also limited in what it can put back into a larger battery system unless you have multiple panels.