Hi there! It sounds like you're going about it the correct way, however I do not see Mangy Wolf on the list of pets that can teach you a Bite rank, are you certain you tamed the right creature?
https://www.wow-petopia.com/classic/abilities.php#bite
If you visit the Bite section in pet abilities, it will list the specific beasts that will know Bite rank 2, which are the following:
Webwood Silkspinner (Spider, 8-9, Teldrassil)
Starving Winter Wolf (Wolf, 8-9, Dun Morogh)
Vicious Night Web Spider (Spider, 9-10, Tirisfal Glades)
Prairie Wolf Alpha (Wolf, 9-10, Mulgore)
Prowler (Wolf, 9-10, Elwynn Forest)
Coyote (Wolf, 10-11, Westfall)
Forest Lurker (Spider, 10-11, Loch Modan)
Timber (Wolf, 10, Dun Morogh)
Giant Webwood Spider (Spider, 10-11, Teldrassil)
Worg (Wolf, 10-11, Silverpine Forest)
Coyote Packleader (Wolf, 11-12, Westfall)
Lady Sathrah (Spider, 12, Teldrassil)
Loch Crocolisk (Crocolisk, 14-15, Loch Modan)
Oasis Snapjaw (Turtle, 15-16, The Barrens)
Tarantula (Spider, 15-16, Redridge Mountains)
If you grab one of those and then fight alongside it for a bit it should hopefully give you the abilitiy!