Battery and inverter.
Replace the battery.
For testing the inverter, use a true RMS DMM and watch it while drawing an actual load. You ~SHOULD~ be able to see the voltage fluctuating. If your fluctuation is greater than 10%, it is out of UL specs and should be replaced.
If you are familiar with a Wiggy, you could also use that to test, as it is an actual draw down meter. But I am reluctant to recommend it, as bad things can happen if you aren't familiar with how one does and doesn't work (intermittent cycle rating).
Good luck. Trouble shooting can be a b!tch.
*Also, while it probably, goes w/o saying, check the Neutral and Ground connections on the inverter.
