I'm not sure that that's a cigarette lighter. Probably a power port. A cigarette lighter will carry about 15 amps. A power port only 5 amps without over heating. If the OP is running the inverter with a 50w 120volt draw, the 12 volt wall port/plug will see 4.8 amps at 100% efficiency. My guess is that he's having voltage problems due to excessive heat. It could be the inverter is problematic or just responding to the erratic voltage.
2000 Cherokee 29BH with 6V batteries, LED lights & 400 watts of solar power, flipped axles and raised. 2007 Tundra 5.7L DC-LB with lots of mods. C-co, 8/158th AVN Maint.