Has anyone mentioned Trent / MyGeekShow here yet? Sadly he has removed all his youtube videos so you can't go back and watch them. :-( :-( :-( Back in the day it was one of my favorite channels. He was literally going through the process of engineering a plane (and power systems and everything else) to do a West coast -> East coast journey. From my recollection he made it all the way across CA into NV before calling it due to the accumulation of issues. It was totally fascinating though.
He built a flying wing that could cruise close to 70mph (i.e. freeway speeds) and had enough battery to fly (I don't recall how long, but) for substantial legs. It ended up being high wing loading and extremely challenging to launch successfully and land safely. (One of the issues with using a flying wing for this type of project.)
One comment: as far as I know, it's totally fine to manually fly an RC airplane from a moving chase vehicle (obviously as a passenger so you can focus on your plane.) I've never tried it ... I would worry about trees or bridges momentarily causing me to lose sight of my plane, and a convertible with the top down would be nice for this, but nothing I own. If you are in a southern state, you can probably just rig a lazyboy chair in the back of a pickup with a cooler of beer. I wouldn't think there would be a problem to have an AP assist with basic altitude and heading hold as you fly. I don't think part 107 would have to come into play if you did this entirely for recreation. That said, if you find yourself in a situation where you are trying to explain the nuances of FAA regs and AMA rules to a county sheriff or angry property owner, I don't know ... good luck with that part.
From the technical side, I support the suggestions to choose some sort of motor glider (with a tail.) Find the minimum throttle setting where you still have good control of the plane (just above the point where the plane really starts to plow and controls get mushy.) You will crawl along, but that will give you the best time aloft. If you were really serious, you would go up and do some flight testing and data collection with your specific setup to figure out the optimal speed or throttle setting to get the best miles per gallon. You can squeeze out a few percent of efficiency with good choices for battery, motor (that is most efficient at your target speed), prop, airfoil, etc. But those effects will be relatively small. But over a long cross country journey they could add up to a lot fewer (or more) stops and a lot fewer or more hours in the air to get there.
You want an airplane you can easily hand launch and belly into grass or weeds just about anywhere on the side of the road or any rough field. I am picturing something like the venerable skywalker (t-tail, fat sorta glider.) It is not efficient, but you will quickly come to appreciate how easy it is to launch and land anywhere.
Oh, my understanding is the prevailing winds in the USA tend to average more west -> east, so you could save some going that direction instead of east -> west.
If you do use any kind of autonomous systems, make sure it can handle taking off at some altitude and then flying and landing at a much lower altitude. Many AP systems assume ground level is where you take off and don't like to let you fly below that. They aren't designed for long cross country flights through variable terrain.