Thanks for all the help, everyone. I'm sure the info on the manual method will come in handy at some point, but I eventually managed to get App::Fatpacker to work on Spreadsheet::WriteExcel.

I manually created a lib folder and copied the module's folder structure into it before running the last App::Fatpacker command and it worked. I'm not sure why it didn't work out of the box, but I'll take the end result.