Check continuity from the light sockets (remove the bulbs first) to the BCM connector. Pull up the schematic on
http://schwarttzy.com/h3-hummer/ to see what wires and terminals to check. Every interior light feeds through the BCM, both positive and ground. The only exception is the grounds for the map lights (which are switched by the buttons). Keep in mind that the floor lamps are on a different wiring circuit than the overhead and vanity lights are. I'm not sure if they are on different circuits inside the BCM as well.
Test voltages with the bulbs removed. Test the ohms of the bulbs, and test the bulbs using jumper wires directly to a power source to ensure there are no issues with the bulbs themselves. Make sure the light socket contacts are clean, the LED bulbs seem to be sensitive to them being dirty.
Put the incandescent bulbs back in like suggested already and see if those work. Maybe do this first and then if they still don't work do the tests I already mentioned.
If the incandescent bulbs work then verify polarity of the LED bulbs, just because you think it's not that doesn't mean you shouldn't bother trying it.
Make sure the voltmeter you are using is not one of the ones they give away free at Harbor Freight, the leads will break just by looking at them wrong, which is a shame since the voltmeter itself is perfectly fine. Spend some money on a decent meter with decent leads, $20 is plenty. I picked up a Uni-T UT210E clamp meter for $46 off Amazon. It does up to 100A AC and DC amps with the clamp (I verified this with testing and research), and up to 600v True RMS AC and DC.
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