[jseutter]

It should be fine to try. Some devices dual purpose the 120 volt transformer to both charge the battery and provide 120 VAC, but those have internal switching so the 120 VAC will simply not work when charging. Given that your device charges using a wall wart, it won't be this type of device because the charging transformer is built into the wall wart. Having a lead-acid battery the charge from the wall wart likely goes to a common power rail that connects to both the battery and the inverter. About the only bad thing that could happen is that a crappy wall wart might try to supply too much power and overheat. I would check if the wall wart is getting excessively hot when running your test.

Also, I took a look at the user manual for the device and it doesn't mention not to try what you are thinking, so it should be fine.