When he flips on the trigger of not casting a spell, can he technically flip twice, meaning that he would turn into a werewolf and then back again, or does he remain as a werewolf permanently?
Illusionist's Bracers only copy activated abilities.
112.3b Activated abilities have a cost and an effect. They are written as “[Cost]: [Effect.] [Activation instructions (if any).]”
Most transform cards, like werewolves and Delver of Secrets, have triggered abilities. Illusionist's Bracers have no effect on them. They will, however, double the activated abilities of Bloodline Keeper, causing it to flip twice when it transforms.
Mayor of Avabruck doesn't have any activated abilities, so I fear this is a bit of a nonbo. Equipping him with Illusionist's Bracers won't usually do much...
Something like Civilized Scholar may reap you greater rewards. If you activate HIS ability you will indeed get to use it twice (and transform to Homicidal Scholar and back again straight away.)