As far as battery charging while driving, it "depends".
A motor-home has large heavy-duty cables allowing in the engine alternator to supply more amps to recharge the batteries.
A trailer most likely uses a 7-pin connecter with small wires. That will allow only 5 to 10 amps of current for every hour of driving. Not enough to meet your expectations.
Unless your old motor-home has been modernized and upgraded (highly unlikely) you will have problems. You need to test your usage out by camping near home to see how your current set-up works.
The worst thing someone can do is head out camping like you intended and find out the batteries are dead the very first morning you wake up camping.