I am new to this website, but experienced in 12 V systems. SInce there was no pos re:this problem, I am assuming it's still there.
Stored for 3-4 weeks... The coach/motor/ electrical system has current drains even when "master switches" were off. Remember: most have a key fob tat may unlock the doors, etc. There has to be a receiver and electronics somewhere that can't be turned off for it to work. The batteries are being drained and the time spent running the engine is not enough. If your alternator is set to say, 14.3 V, the differential of voltage between the battery's INTERNAL voltage sets current, which starts out high and tapers off. However, the differential may start at 2 volts. After a short while, the differential is ! V and current is substantially less, then 1/2V, etc. I one did an experiment. Had a battery of 100 AH discharged to near 10% capacity, with a constat 14.3 V source that had 100 Amp capacity. Ignoring the tapered current, it supposed to take an hour, right?. Wrong. Battery charging is only about 70% so you must put in about 130 Ah to get 90 AH charged. Moreover, the current tapering, our lab test showed about 4 hrs of charging to attain 95% capacity.
This was done on the deep discharge regular batteries of 2006. Of course, the lithium cells of today are better, but recharging is recharging.
The above advice is right on about solar panels, but use a regulator made for this. Putting current into a charged batter is not good for it
Hope this helps.
Note: in cold weather batteries lose power as compared to warm.