Anthropologist here by study.

It'd be inaccurate to break nomadic and semi-nomadic societies down by what they gathered because what they gathered was a direct consequence of what was available. In some capacity all pre-revolution societies engaged in some level of hunting, gathering, pastoral and horticultural. The only reason northern tribes who settled in predominantly arctic territory, such as the Inuit, may have forgone pastoral food sources is simply because they're hard to maintain in such conditions. A commonly offered explanation for why humans settled down into static societies in the first place was that it became a point of necessity due to population size versus sustainability of nomadic food sources.

To be blunt you'd be better suited identifying them by time period, location, or survival strategy (nomadic, semi-nomadic, sea-faring, ect) since their diet is almost entirely at the whims of the weather and altitude. We know that the Native Americans living in the Willamette river valley in Oregon, despite being famous for the massive historic volumes of salmon in the Columbia and the Willamette, were also prodigious harvesters of camus bulbs when they were in season. And there was another tribe in the high desert that'd literally take whole bison and dump them in local area lakes before they froze over so that the following Spring when they're back in the area, they could just recover the frozen body and have an easy source of protein. Somehow I think "proto-refrigeration" doesn't roll off the tongue or sound sensible in a linguistics paper.

In order for food sources to become relevant, you'd need to establish why it even pertains to language. That being said because food sources often become very relevant in the context of wealth transfers such as dowries and barter, I guess you could have some wiggle room if you can argue persuasively.

Unfortunately linguistics, especially of pre-Columbian American groups, is also going to be a black hole. I have to imagine it's not much better anywhere else. To this day there's even ancient Greek languages and alphabets we don't entirely understand. Have you talked to your professor about this? I tended to find in my undergraduate that every time I was overwhelmed by the project I was concocting in my head talking to my professor either sorted things neatly, or completely leveled the playing field when they said, "no, don't do that, I've seen enough idiot undergrads do that, do this instead." Because it does sound like you're trying real hard to bite off more than you can chew.

Oh, and you may want to be more specific on "Lakota." Apparently it can ruffle feathers kind of like how a lot of the region's native Americans don't like being identified as Sioux either.