No problem leaving it on one battery then swapping it to the other side
So, if I use a normal 12v auto charger I must in-hook the opposite side battery vs using the 24v charger I can keep it hooked up and if I want to charge both at the same time I need to run leads to both battery's?
If you want to us a 12 volt charger to charge both batteries at the same time you MUST disconnect the batteries from all the generator cables.
Then you can connect the two 12 volt batteries in "parallel". IE connect one battery's + to the other battery's +, then the one battery's - to the other battery's -. Then hook up your 12 volts battery charger connecting the chargers + clip to battery 1's + terminal and then connecting the chargers - clip to battery 2's - terminal.
When you need to use the generator disconnect the chargers, all the leads and then reconnect the 3 generator cables.
It's much simpler if you just use a single 24 volts charger or two 12 volt chargers (one for each battery).
I use a pair of 12 volt Battery Tender's to keep my 802's batteries topped off.