(See Updates at end of this post)
For $reasons, I need to codesign an old Director projector that we can no longer re-publish (no access to original source code or to Di
If the app can't find its external resources when being run normally from the Finder, that sounds like a result of Gatekeeper Path Randomization. The OS moves the app to a hidden read-only location when it's run, and it can't find the resources.
I don't think that signing the app binary itself will fix this problem and prevent the OS from doing path randomization. The user either needs to move the application to a different directory after extracting it, or you need to distribute the application inside a disk image that has been signed with your Developer ID certificate. DropDMG (linked from the post above) can create signed disk images, that might be worth a quick try.