I'll try this.
The way I have to resume downloads in Firefox is
1. Move the .part file to a temp file
2. Start the download again; a new .part file gets created
3. Pause the download
4. Move the temp file to the .part file, overwriting it
5. Unpause the download; it resumes from where the old .part file stopped
This is only necessary if a "one-time" download link gets interrupted part-way through, eg. by a flaky connection. Attempting to resume a one-time link like normal gets (I assume) a 400 response and FF kills the .part file.
One caveat with that trick: if the server picked a different file or the link now points at a new version, Firefox will happily splice the old .part onto the new bytes and you get a corrupt file with no error. Resume requests only check the range, so unless the server returns an ETag or Last-Modified that matches, nothing guarantees the two halves belong together. Worth checking a hash afterwards if the source publishes one.