While I'm unsure whether Emerald restricts the release of the last 2 mons (never tested it myself), it sounds like Poke J's tests were pretty conclusive.
As for the mandatory last doubles 7th badge, I imagine this happens when you don't restrict the release of the last 2.
(AFAIK, RS only restricts the release of the last mon, and the user in the video placed HM mons in daycare, and the Abra that was used to Teleport back to Mossdeep could be released with no issues [not HM, not last mon, cause 2 in daycare])