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 https://rpg.stackexchange.com/questions/53188/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 https://rpg.stackexchange.com/questions/55358/barbarian-rage-and-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!