After doing a bit more digging I'm going to change my answer to

Yes

You already quoted part of the Shifting text which is relevant. According to this question Can a Druid/Monk make unarmed strikes while Wild Shaped? a Wild Shaped creature can make an Unarmed strike , which the shifting feature for a Longtooth specifically calls for.

Also taking into consideration How does Rage interact with Wild Shape?, Barbarians can rage before Wild Shaping and still retain that rage.

I said earlier that I think it makes sense for flavour. But it now looks like the rules also back that up. Happy Shifting!

#Yes

You already quoted part of the Shifting text which is relevant. According to this question Can a Druid/Monk make unarmed strikes while Wild Shaped? a Wild Shaped creature can make an Unarmed strike , which the shifting feature for a Longtooth specifically calls for.

Also taking into consideration How does Rage interact with Wild Shape?, Barbarians can rage before Wild Shaping and still retain that rage.

I said in an earlier answer that I think it makes sense for flavour. But it now looks like the rules also back that up. Happy Shifting!