Approach shoes are good to wear car-to-car. Another option is to carry rock shoes and change for the slide. Rock shoes are not very big or heavy, and that option lets you wear your favorite trail foot wear on the trail, and the very specialized rock shoes on the rock.
If you have not worn rock shoes on a slide before, you don't have to worry too much about getting a good pair. Any rock shoe will feel fantastic on a slide, if you are used to boots or trail runners. My recommendation would be cheap, loose, and comfortable. A nice pair of used rental shoes, a half size large so they're confortable with socks, would be my choice. I keep my stretched out, beat up old rock shoes for use as "slide shoes."
A caveat: I would not use the closely fitting rock shoes you use for vertical climbing. Slide climbing, as you know, is basically walking, and involves a lot more foot flexion than vertical climbing. Your toes and heels may hurt, and you may stretch out your good shoes, if you wear them on a slide climb.
TCD