Those string with embedded lights aren't quite a normal light bulb, so I'd try the inverter with a standard table lamp before I decided that there was a problem with it.
If your 12v outlet is low on voltage, your 120 VAC output will also be low and cause a brown-out. Being under-wired (low amperage) should just blow a fuse or burn up the 12v wiring. But an inadequate outlet could cause both low voltage and low amps, so its possible that's the problem. Try it somewhere else, e.g. the car. Better yet, if its an SUV it may have a utility outlet instead of the typical cirgarette lighter socket. Most car 12V outlets are wired for 10 amps, though some may be 15A.
As for powering battery chargers from modified sine inverters, your mileage may vary. Some chargers get upset with them and buzz loudly or just quit, some work but get rather warm and others work just fine. I've lost one and don't care to risk another. I also had an X10 remote control system set up in our coach to control decorative lighting on our site and all the X10 relays and controllers burned out when I ran them on our 2000 watt inverter. EVERY one of them. They weren't even in active use, but they were plugged in and I realized something was wrong when I smelled one of them cooking... The charger for my Bosch drill/driver was one of those that buzzed loudly enough that I was afraid to continue charging it - had to wait til we had the genset on. On the other hand, all the laptop computers we've used have been happy on modified sine. They probably have better power management circuitry.