Never heard of the battery being affected by MSW. Usually the charger itself craps out in my experience with a porter cable Nicad and a Craftsman Nicad charger.
But Li-ion is a whole different ball game as the cells are individually monitored as over or undercharging them can be dangerous.
I bought a 400 watt PSW Wagan, basically just to power my Makita Li-ion charger as I was unwilling to risk it to MSW.
I've basically Disconnected and removed my 800 watt Coleman MSW( needed the space) as the Wagan has half the standby current, makes no noise providing under 150 watts, and I rarely need that extra 400 watts. Perhaps for the angle grinder, but I still have the ring terminals insulated and ready to be reattached if I need 800 watts.
My battery monitor shows that the Makita 15 minute charger, through the Wagan, pulls 17 amps max. No ciggy plug is going to handle that, but I imaging 30 minute or 1 hour chargers consume considerably less at max draw.