Caribou’s close relatives, the reindeer, were semi-domesticated in Northern Europe, and they are similar. Bighorn sheep, while not as conducive to domesticity as domestic sheep, still have social structures conducive to domesticity based on what I’ve read. Bobcats and American wolves could certainly be domesticated, and elk seem possible if there is a pre existing caribou domestication event.
Elk and caribou seem to overlap in their roles for humans. What does the elk offer which the caribou doesn't already offer? The caribou seems easier to domesticate, since it has predictable migration patterns and we know it was domesticated by nearby Siberian groups (including the Kets, very distant linguistic relatives of Athabaskan-speaking Indians). Although elk are larger, Alaskan caribou are sizable so would be sufficient for a culture with no domesticates bigger than a dog. Moose would be more logical, at least if the civilisation domesticating them is farming aquatic plants (i.e. wapato, wild rice, etc.).
I think a key factor with caribou domestication is giving the Athabaskans or whoever will domesticate it a reason to intensify their use of caribou. Like in my TL I started with the cultural idea of encouraging subarctic plants and lichen to feed more caribou while feeding humans, which eventually expands into controlling which caribou can breed and then using individual caribou to carry packs and other goods, which is a key innovation since being able to carry more goods is huge for hunter-gatherer cultures. In my TL I extended this to the final step in domestication by having caribou be used to help in the construction of earthworks for prestige of powerful leaders which leads to the cultural idea of using these earthworks for irrigation and thus more food for caribou and humans.
I'll note that either caribou and elks are vulnerable to parasites and diseases carried by various deer species. Caribou especially, and white-tailed deer are a major carrier. Elk seem to have evolved some resistance although were easily extirpated from their native range, while caribou have much less resistance.
Bobcats are logical, since they were eaten for meat and used for pelts, as were lynx, although they are rather big to fill the role cats did, so they might have to share their role of pest control with mustelids or smaller dogs.
Muskox are also a good domesticate, since their fur makes a good trade good and they can produce a lot of meat, although they have limited range since they can't tolerate wetter climates.
These two videos by CGP Grey should hopefully help. Something I can see cropping up in your timeline though is the rise of an Americapox when the New and Old Worlds collide, as more domestication = more concentration of people with animals = more plagues and diseases.
Although virus and bacteria mutation produce a lot of possibilities, we should recall that the majority of zoonoses won't be transmitted between people. IIRC cattle and pigs have produced the majority of diseases which have spread to humans. If American domesticates are limited to waterfowl (geese, ducks), cervids (elk, deer, caribou), and goats/sheep, it isn't likely they'll produce many serious plagues, although no doubt they'll spread unique forms of avian influenza (which given Mesoamerican domestication of the Muscovy duck, I'm curious of whether the Mesoamericans suffered from influenza, although perhaps their influenza was hard to spread, like the H5N1 strain has been)--avian influenza is the easiest American disease to spread, since "Ameriflu" will be very divergent to outside influenza diseases (in terms of immune response) and could easily cause a Spanish flu-level pandemic, although 5-10% death from a pandemic OTL is under-reported in records and not too unusual so may go unnoticed compared to the usual ravage of disease. I think you'll find the handlers of domestic animals--they'll tend to be lower-class people in the service of nobles who own the animals--will suffer from zoonotic diseases at times, but these diseases will not evolve to spread between people. Major animal diseases (like which will plague caribou or elk) will stay limited to animal herds. Other viruses lurking in the Americas (hantaviruses, etc.) may be reliant on certain conditions to cause serious epidemics (i.e. cocoliztli) and would be hard to make endemic diseases.
This isn't to say the Americas will never produce any dangerous diseases, but it isn't inevitable that domestication of more animals means more diseases which are equivalent to what the Old World has.
I’ve actually considered the bison, it probably just isn’t naturally prone to domestication like, say, a bighorn sheep and would be significantly more difficult. It would probably be the last domesticated animal.
I don't see how a bighorn sheep is prone to domestication, it's social structure isn't well-suited. Mountain goats or Dall sheep are better suited for it.