I suppose if the decision was reached mutually and you both want to establish a new mode of “just friends,” I’d say do something that just friends commonly do. Sports, for instance, keep you occupied without demanding much emotional engagement. Maybe take lessons at a climbing wall? But if you’re talking about a breakup date, it’s going to suck no matter where you’ve gone, so why sweat the location, other than making sure there are enough walls to provide some privacy. There will be tears.
For a breakup? Huh. Sounds creepy. If you’re gonna break up, at least do it somewhere warm with sugary treats.