I would be using 1/0 OFC wire if that was me. That should solve it getting hot. Length also matters, the shorter the better, but not as important
You didn't say what size fuse is blowing.
* Your fuses needs to be the same size or larger then the load/fuse sizes downstream.
* Your wire needs to be capable of handling more than the fuse.
(The fuse is you intentionally adding in a weak point. So that if something goes wrong it is in an controlled location, that can be easily fixed).
I personally wouldn't do more than 200-150 amps on 2ga, and that's with OFC wire. Copper Clad Aluminum is closer to 150-100amp.
The other thing that might help:
3,000 watts / 12v (engine off) = 250 amps
3,000 watts / 14.4v (running ) = 208 amps. You will have a lower draw when using the inverter if your engine is running. It's a 20% difference, and might be the difference in you having it work long enough for you.
If you are already running the engine (at 14.4v) when using the inverter, then I would guess you are using a 200 amp fuse, maybe even smaller. That inverter needs at least a 250 amp fuse, and you need wire larger than 2g so that the fuse remains the weak point in that circuit.
I know for close to 20 years, GM has also been using external voltage regulators on pickups. Instead of the 14.4v being controlled by the alternator, it's controlled by the ECU. It lowers the voltage to reduce drag, and increase mpg. If your 2018 Dodge does something like this, perhaps even running/at idle, it's reducing your voltage which is increasing the draw. No clue if your Dodge does this, but it could be dropping below 12v, which would draw even more than 250 amps, and could blow a 250 amp fuse. Could also explain why it blows after 5 minutes.