There's a couple things I am curious about. Are you still running the original battery? If so, that might be the problem. Many of us had issues with the original OEM lead acid batteries causing glitches and issues on their way out. Everything is so sensitive to current consistency, the lead acid batteries would cause problems but still start the truck fine.
If it's the original battery, I'll tell you what I tell everyone... Walmart, 4 year warranty AGM battery for $140 will provide much more smooth power. AND, when you take the old battery out, there's a plastic piece at the back of the battery tray towards the cab, POP THAT OUT! Then you can fit a larger H8 battery in there for the same price.
My next questions would be, how did you tap power for the inverter? We have Ram 1500's in our town as police cars since we're country. Fiends with one of the PD and he told me when I was talking about installing a couple things, I HAD TO TAP THE CIGARETTE LIGHTER for power. He said the fuse box under the steering wheel and the one under the hood would have problems with anything of heavier load. It should pull power from a secondary battery or a tap off the dash. I got one of these... never installed it since the onboard inverter handles the fridge cooler but, might be worth some more research.
https://infotainment.com/shop/interior-accessories/wiring/u-intezdc/