Fixed your leaking vapor barrier? Don't forget to lift the carpet.
BMW uses a high-density foam backing that is bonded to the E39 carpet. It's a great noise insulator, but it also acts as a giant diaper when the vapor barriers leak.
So, if your rear carpet feels damp to the touch, it's actually saturated with a shocking amount of water that is held within the foam.
Once water gets absorbed into the foam, it's impossible to dry without lifting the carpet.
There are several bundles of electrical wires under the carpet, so I think it's beneficial to thoroughly dry the foam backing.
Start by removing the back seat. Then the rocker trim. It uses a 10mm bolt with 2 push-in fasteners (be gentle.. this piece is brittle)
Then I popped off the bottom of the B pillar cover. This allows the carpet to be folded back and lifted up.
I've spent over 2 weeks placing absorbent ShamWow towels (made in Germany. You know the Germans make good stuff) under the carpet overnight, wringing the excess water, letting the towels dry, and repeating the process again and again.
I've pulled out so much water that it makes me wonder how many electrical problems could be attributed to wet carpets.