Less than 2 hours after emailing BBEdit Support I have a helpful response which has solved my problem:
The OS will generally quarantine files that have been downloaded, but may do so for other reasons; there are various tools for correcting this, but the most straightforward is probably the `xattr` command line tool.
The following Terminal command will remove quarantine from any given file (use the actual File path rather than "/path/to/file"):
xattr -d com.apple.quarantine /path/to/file
and once you have done so, this issue should not recur.
