For releases there is a checkbox to mark releases that indeed have no barcode (as opposed to the data just missing / being unknown).

This flag is not available in the webservice. Take for example the release https://musicbrainz.org/release/2c59ca24-d4ed-4265-94cf-14c65c62daa9/ . The web service response (https://musicbrainz.org/ws/2/release/2c59ca24-d4ed-4265-94cf-14c65c62daa9?inc=aliases%2Bartist-credits%2Blabels%2Bdiscids%2Brecordings&fmt=json) provides the barcode as an empty string, but from the data it is not clear whether barcode is just empty or marked as non-existing.

Maybe there could be a boolean "no-barcode" field. This info is a requirement for PICARD-2799