Actually battery chargers can be problematic on modified sine wave(modified square wave) . So it might not work on anything but, pure sine wave. A pure sine wave inverter can give you cleaner power than what you find from the utility sometimes. Then you get into high frequency designs versus the low frequency designs. Unfortunately smaller pure sine wave units that are low frequency designs are hard to find, and then if you do somewhat cost prohibitive.
You can usually tell a high frequencies unit by it low cost, low weight, (no transformer) and they almost always list their surge output as twice the available power, what they don't usually say is how the time component in that surge spec is very low less than a second. A low frequency unit will have a surge less than double, but it will be a 5 seconds and only drops a little more going to 10.
The high frequency units generally run hotter, have a higher chip/component count thus don't seam to last at long. I would try to mount them vertically, although be careful if your a door slammer.
So as far as some suggestions, price not a worry
Sensata Dimensions 12/400N 400 Watt 12 Volt Low Frequency 400 Watt 12 Volt Heavy Duty Pure Sine Inverter
or you could drop down to a high frequency units such as Magnum Dimensions CSW412 400 Watt 12 Volt Compact Pure Sine Wave Inverter
or if for your Milwaukee tools try a direct 12 volt charger