Turns out it is 1930s, which gets rid of the continuity problem.
However, the scene in the movies shows very briefly some Quileutes with animal skins on their heads and (apparently) draped over their shoulders, an effect similar to DC's Flying Fox.
In my opinion, this type of dress for Natives might fly in the mid-19th century, but seems out-of-place in the continental US in the mid-20th century. Agree or not?
(The Quileute-wolf origin story is revisited in the later "Twilight" books, and will likely be revisited in the future movies. The movie makers have a chance to present it in the movie in a way that is true to Quileute legends without wrecking the story.)
Some evidence for this assertion:
Indians of the Northwest, 19th century:
Chilkat dancers, 1895:
Quileute children, 1887:
To me, this flashback kind of encapsulates the whole Twilight mentality. The Quileutes were wolf people long ago. Dressed in their furry cloaks, they were wolf people 80 years ago. They're wolf people today. And with Jacob imprinting on Renesmee and (eventually) having a litter of puppies with her, another generation of wolf people is assured.
As for your other question...yes, Twilight's producers could make the legends more authentic, but will they? I'd say their willingness to replace Taylor Lautner with another non-Native is a bad sign. I'm guessing they want more sexy melodrama, not less.
For more on the subject, see Quileute Werewolves in Twilight.
Below: "I'm a wolf-boy! I don't need clothes!"