At first I thought this meant that I just had to keep the copyright notice inside the source files, but apparently it means I have to include their notice somewhere as a separate file along with my game(s).

The accepted consensus seems to be that what you described is zlib/libpng license which is even more permissive than MIT.

With MIT you have to credit the author and mention the license in some way. The 'include' part might be ambiguous, some people consider link or name of license is enough, some want entire license pasted or attached to game as text file. Your best option is really asking the author about the form of crediting and possible exception for you. MIT license is very friendly to commercial use so he might be more than happy to help you.