If you're looking for some subspecies of Felidae that might "make it", Viderridae are your best bet, as they are omnivorous to begin with. They're the least specialized family of Felidae, and the most primitive, which might not be what you're looking for in your worldbuilding.
As for "the big cats", I'd say possible, but unlikely.
While not exactly everything, but many things are possible in nature, so I won't say "impossible".
However (emphasis mine):
...a sudden, dramatic drop in temperature wiped out half of all plant and animal species... ...would it be possible for any latest-Miocene felid species as big as or smaller than a puma to transition from hypercarnivory to hypocarnivory?
The bigger, more advanced families of Felidae are very much specialized as (ambush) hunters, much more so than Caniformia (dogs, bears, seals, racoons, weasels, and that rather special red panda). From ambush hunter to plant scrounger with the occassional prey of opportunity is a long way to go. It's not just the digestive tract and the teeth, it's being good enough at it to find a niche among the much more effective plant eathers.
Evolution works rather slowly. Which is fine if adapting to slowly changing climate / environment like at the end of the Miocene... But you explicitly mentioned a sudden and dramatic change, within one generation or two or even due to a singular catastrophic event (?).
A transition would take many generations of evolutionary pressure, but those hypercarnivores are starving today. It would be much more likely that big hypercarnivores would get displaced by species more fit to cope with the sudden change.
You could come up with mitigating factors, which would allow your hypercarnivore Felidae to survive, likely in much reduced population numbers and / or species variety, and indeed "make it" into this new age of your world.
If however your ultimate idea is having those felidae be / become once again a "ruling" family of any description, that would take quite some time, and lots of mitigating factors.