Congrats on passing your test! I am still running on my CDL learners from Virginia and will do the driving test this spring. I'll be sure to "never" look straight ahead
and always wildly throw my head from side to side like watching a tennis match on steroids!
Your OP said the inverter "display" was on (though dimly) which I might suggest infers the inverter is still on in some way and drawing some juice. The ME display is taking some load itself.
In my opinion, 3amps is not an outrageous parasitic load though it will deplete your batteries in no time.
I'd look around for a hidden (basement?) light on, or some other rogue load. Maybe pull some key fuses while you monitor the load to see what is going on. That should help isolate the drain.