Hi,
I have never had a problem with my charger overheating while running from my modified sine wave inverter. I guess that the manufacture has nothing to lose by printing "Do not use with inverter" and if someone had used a new charger on a bad style inverter, then the tool manufacture might have had warranty issues.
I only know that the laser printers from the 90's could not be used on modified sine inverters, that might have changed in the past few years though. I have used a ink jet printer on my MSW inverter many times, with no damage.
The only thing I did damage on MSW Trace 1500 watt inverter is my microwave. The replacement is only run while on shore or generator power now.
Fred.