AH-HA! I figured out what the problem was.
When I initially went to apply the 3.7 patch, I forgot to change the install path to {x:\Program Files (x86)\SEGA\Medieval II Total War}. It's that little (x86) there that was the problem. See, under all 64-bit versions of Windows, programs that would be installed to a {x:\Program Files} folder need to instead be installed to a {x:\Program Files (x86)} folder, and the Patch installer does not take this into account.
Please update your patch installer to remedy this.