I'm hoping that you can test and avoid going into the wiring across the top, etc. That's assuming and dangerous but the wire seems like the least likely to be the problem??
I have no proof and never see it explained but looking at lots of drawings, it seems the ID like TR1/TR2 and maybe a TR3 are meant to tell us it is a multi-legged/branched wire with several ends. Maybe TR1 at one light, TR2 at the other and maybe TR at the resister would sound right, except that last drawing above also shows TR1 at that resister, so still have something to figure out on that part.
And that is true for me and LED as I have not had enoguh trouble and no training on them other than just switching out light bulbs at the house and some installing to replace in the RV, so that leaves me in the dark.
I was thinking what I might do and if there seems nothing wrong at the resister, I might think about this way to go.
One is the question of them being soldered in. Is that a problem to remove the joint and resolder?
If you can get them out without lots of trouble, maybe a switch of a resister for another if the testing still leaves them looking good? Sounds like a lot of fuss to swap out resisters if you can just get one loose and test. At least they are not a high tech black box that we can't know what inside!
I don't know if this fits or not. Rather than remove the seat, can you possibly reach under, test resistance from the back to the resister by putting a jumper clip to ground on the near side of the rester and then compare resistance to the jumper on the other end of the resister? Idea being rsitance to ground on first side and then resistance to ground on second side being ten ohms more says the resister is actually ten ohms or close. Any sense to that?
But then moving beyond the resister, I might look at pulling both led bulbs out at the back so that I'm guessing it would then leave just wire from front to back with a branch going to the second bulb. Chjeck that guess before trusting!!!
But if true, I would want to look at the wire for having no voltage under any condition like the other lights on, etc. . Just looking for wire with not crosses to other wires?
If finding none, then maybe a helper to let you look for good connection from front to back by them putting a jumper to ground as you read it.
Sorting out what we mightguess is there?
Resisters? IF they look good and no stray contact like loose wire ends,etc. and they resister measures good? We kind of HAVE to call them good.
Wire to the back? If you have it open at the bulbs and see no voltage and see them good continuity, we call the wire good?
New thought? How about the ground at each bulb in back? We think battery is getting there because it lights for a short time, but ground can be funny and do weird things, so maybe a good time to check it at both bulbs?
I don't think of LED (diodes?) as passing current the wrong way. They are definitely polarity sensitive. That's been my major experience as I can put way more than half of them of those little wedge types in backwards, without even trying!
If all that checks, I'm getting low on ideas! Four days to find the problem and two minutes to fix it!