Unless you have confirmation from the game developer that the Pause key will actually work where you're hoping it will, I really don't think it will. These days the Pause key is basically limited to halting the boot process, and even that use case is rapidly disappearing given how quickly modern UEFI systems boot, which is why the key itself is disappearing.
But if you want to keep trying, the simplest solution is probably to just plug in an external keyboard that has that key, either by borrowing one or just buying a cheap $10 one since it sounds like you've probably spent more than $10 worth of time on this by now. Otherwise, AutoHotKey is a popular tool that can do all sort of key mapping and macro wizardry. It's not especially simple to use, but it does have great support in their forums, so if you Google "AutoHotKey remap Pause", chances are you'll find a post that has instructions. But I'm also wondering if the Pause key might be a special lower-level command that might not be able to be remapped in the first place, in which case even AutoHotKey may not be able to help, which takes me back to borrowing/buying a cheap keyboard.