Okay I managed to figure out the tab. What I thought were two lips that held the whole thing together ended up being a place to slide a screwdriver in.
I got two flat head jewelers screwdrivers to pry the thing up and managed to fit them straight through the lifted tab - to keep it pulled.
Pushing the wire out took more force than I expected, but it came out very cleanly and without much fuss. Wrapped it in tape sticky-side out, and then wrapped it in more sticky side in. And then taped it to the firewall insulation.
I also noticed that since I've done this, already on my way home at two points where people always pull out in front of me - today they didn't. This tentatively confirms my suspicions that DRLs lower respect on the road (as people rarely pulled out in front of me in my old car).