I'm trying to get Dungeon Hack to launch on Linux, via Heroic Game Launcher, but can't get it to start up. I'm running Debian/Testing, I do have Steam & Proton installed. When I click Play, it seems to try and start up, but nothing happens, no window starts up, process seems to start and die extremely quickly.

I don't know how experienced you are with Linux, but mostly I don't know how experienced you are with having to possibly copy files and then setting up DOSBox properly.
But, my suggestion is to use your distro package manager to download DOSBox Staging or DOSBox-X.
Dungeon Hack is a DOS game, and if you mimic how GoG sets up up their included windows version of DOSBox on a Linux native version of DOSBox then you would not need Heroic, or Proton, or Wine.