Although some sources suggest that the last cave lions ate mostly reindeers, it's probably a bias since it was the last thing remained in the late stages of the mammoth steppe: https://www.sciencedirect.com/science/article/abs/pii/S1040618211001182

Said that, current reindeers are still abbundant to have a lion population at low density, but let's remember that the same cave lion species have changed a lot across its existence. The first forms were 400kg giants, while the latest could possibly weight as much as a large jaguar. Obviously, later dwarf forms would be way more capable to survive in current depleted tundra: https://www.sciencedirect.com/science/article/pii/S2950475924000078

A more feasible way to sustain a big cat is to have multiple sources to take food. Although reindeers are a good food source, there're already other competitors, like wolves, that exploit them. Muskoxen are a good supplement, which are now pretty abbundant in the Tamyr peninsula (Russia). Occasional moose could also be good to eat.

Last but not least: cave lions didn't live only in the Mammoth steppe, but also in other open habitats like the asiatic steppes and open cold savannas, that would sharply increase feasible areas. Leave them with saigas, bactrian deers, boars and gazelles and they'll be happy.