Based on your response to Alex L.'s comment, it sounds like FTL is looking for various things in your current working directory, so if you navigate using a filebrowser or using cd, it works. Since the menu doesn't do that, there's something that it can't find.

Try editing the .desktop file so the Exec line looks like this instead:

sh -c "cd /home/andi/.local/share/FTL/ && ./FTL"

That will switch directories first, and should let FTL find everything it needs to start.