The following assumes that you used sufficient gauge power wires (no less than 8 gauge for all that; 4gauge would be better), and scraped the paint off the metal until it was shiny whereever you connected to ground on the body.
Also, when you say the inverter is wired up to the ignition wire, do you mean that is where you are pulling your main +12 going into the inverter? If so that is a Bad idea. First, that wire is too small of a gauge, second, that circuit in the vehicle is not designed to carry that much current.
Options if it still does that and you have properly wired the inverter and amps:
1) you can add a second battery. But, is the power problem is really serious, do #2
2) upgrade your altenator.
However, DO NOT do either #1 or #2 unless FIRST you wire up that inverter directly to the battery using 8gauge wire. I am willing to bet that is your real problem...