Since there are no specific rules for fiends, they use the general rule.
A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours ...
Of course, if you don’t want fiends to sleep to rest in your world, then they don’t.
Consuming a mutagen is basically the same as eating or drinking. You can absolutely do it during a rest.
As you noted, if you use consume Reconstruction within 1 hour of the end of your short rest, its duration will continue after the rest finishes. The same effect occurs with Catnap or Metabolic Control.
Note that Blood Hunter isn't a WotC class, so it is ...
Vampire Spawn Need Sleep
While nearly all creatures with 'undead nature' don't "require air, food, drink, or sleep" (cf. Skeleton, Specter, Wight, Wraith, Zombie, Banshee, Death Tyrant, Crawling Claw, Demilich, Flameskull, Ghost, Lich, Mummy, Revenant, and Shadow), and will o' wisps don't need air, drink, or sleep, for both vampires and ...
The Nightmare spell from pg.316 of the Core Rulebook states that:
Creatures who don't sleep (such as elves, but not half-elves) or dream are immune to this spell.
So either the CRB is stating that elves don't sleep, or they don't dream. I'm guessing because of the Dreamspeaker alternate racial trait that elves sleep but they don't dream.