I had the same problem with my Samlex 150 PSW inverter plugged into the cig outlet on the antenna booster wallplate. Firing up the TV made the inverter's "low voltage" alarm scream. The wiring run to the outlet was too long and too gauge too small to draw any power out of it.
What I did was get a one-circuit 15A transfer switch, wire the inverted directly to the batteries, and use the transfer switch to power the 120v circuit that includes the TV outlet. No more issues.
Write up on my inverter insstall is
here. Nice thing was that all the wiing was done at the panel, I didn't need to run any wires up to the TV area. I did boost the wire gauge between the batteries and the load center to help with charging and have since added a Progressive Dynamics converter.
I don't think buying a new inverter is going to solve anything for you, but I guess you can try...