Back up your company.   Now go to file restore and restore an OLD backup with the form.  When you choose what you want to restore, choose FORMS only, not Company data.   You will get your very own form back. If something goes wrong, you have your backup. 

The issue with this is that I'm only getting the option to load Custom Forms as an individual selection. This could still be worth trying, except there have been changes to others since my last backup and I'd rather not lose them.
If the files saved under Forms were named more directly then I could just copy out the ones I need, but I can't tell which file is associated with each form, so using this backup/restore method I'm losing something either way.

When you open the form to modify it, it has a number at the bottom.  You will see the number when you go to save it! 

I do not see anything that identifies the actual filename for the form as it's stored in Windows. This number does not help.

I called Sage support again and they provided me with the appropriate files.