While there are a handful of possibilities, I would concur with swapping the brake light switch. Hopefully you ordered a genuine MB one, the aftermarket ones are generally not great for this part.

The alternative is to drop several times the cost of the switch to have SDS plugged in and see what's triggering it. Thus, that's why I'd start with the BLS as well.

It may turn out to be wheel sensor, or perhaps the grooves in a hub are impacted with crud, which is not that uncommon. I clean mine very well every time I replace pads, and they are always pretty dirty.

Good luck, I hope it's the BLS. Easy and cheap.