Specializations.
I feel like I should comment on this, since I was the one who picked this specialization in the first place and this is something that concerns us all:
Tooth and Claw is one of the fighting styles for werewolves from Werewolf: The Forsaken. It's a martial art specific to werewolves, focusing on the werewolf's natural weapons and fighting as a wolf. Kinda like Kailindo in Werewolf: the Apocalypse, except without the form shifting part. Which I thought was thematically appropriate for Leaf Storm, since he is a Lupus and thus I named his specialization after that.
I really don't think it's too broad, considering that example specializations for Brawling listed are Wrestling, Boxing, Kung Fu, Karate, Kailindo and some others. It would be pretty much the same thing if your character was specialized in Melee weapons and picked Klaives as his specialization.
By your interpretation, said specialization couldn't be Kung Fu or Karate, but something like "Kung Fu Punch", "Palm Strike", "Karate Kick" or rather specific Street Fighter type moves like "Raging Dragon Spin Kick" or "Hadoken". And Melee couldn't specialize in Klaives, but... I dunno, Silver Klaive Slash? I understand that you don't want us to be uber strong, but limiting it like that seems way more boring to me then having Karate as the specialization, since:
1) You would be pushed to use the "Spinning Dragon Kick" as much as you can, in place of normal attacks, even when it wouldn't make sense and would just feel ridiculous and you would be doing pretty much the same thing over and over. You have a lot more opportunities for fun actions with something like Karate.
and
2) I really don't like something like Gripping Bite as a thematic brawling specialization for a character. That seems more on the level of making your Strength specialization "dealing damage" or rather like a specific combat maneuver. Or a Gift. So instead of having your character be a great wolf warrior or a kung fu master, he is now very good at holding things in his mouth, because... he loves to play fetch?
This is just my opinion on this matter. If you have to limit the specialization, then my suggestion is not to do it like this, but instead talk with Leaf Storm and others about how they would envision their fighting style and when would it be appropriate to count it as a specialization. Like, breaking down that door wouldn't be something I would consider appropriate for the specialization, but if he like jumped at the vampire, gripped his shotgun in his teeth and tried to disarm him like that, then that seems awesomely thematic to me. Limit it as appropriate.
That aside, I should probably pick my specializations already.