I can tell you that I have 4 light fixtures in my slide and they are all fed by the same circuit. True that there are multiple lighting circuits so a blown fuse doesn't leave you in the dark, but typically only one wire runs to the slide.
Before you drop the belly, extend the slide and look under the floor. On mine there is a flexible plastic carrier that holds the single 120v stranded cord and single 12v pair in two separate 1/2" split loom conduits. These are zip tied to the plastic carrier that manages the movement of the wires as the slide extends and retracts. I would look there first, and see what you can find.
Of course yours may be wired differently than mine but the way the industry works, it seems unlikely that they ran two 12v circuits to the slide lighting. But the wire carrier setup is pretty common to all slide outs. How many 12v wires will at least tell you how many circuits service the slide.
If I had to take a SWAG, I would say that fixture #1 is in reality dead, but picking up phantom voltage somewhere so that you get a 13.6 reading with no load, but the slightest load (even from an LED) makes it disappear.
If you can't find the problem, Plan B would be to disconnect the wires from fixture #1 and cap them in the ceiling (allowed with low voltage, but not with 120v) then fish a new wire thru the ceiling from fixture #2 to power fixture #1. Then they will both be on the same circuit anyways. :B