I think in order to understand how they would evolve, you'd need to clarify what purpose they fulfil within your fictional society.
Wolves are persistence predators, and one of the few predators in the world who can keep up with human persistence predation tactics and actually be useful for hunting things. This is one of the major reasons they have been domesticated where a dizzying array of other predators haven't.
Big cats are ambush predators, and as such tend to run out of puff very quickly (but be capable of immense speed and acceleration over a short distance). This is not as complimentary to human hunting characteristics as the lions can't keep up with humans, and humans blow the lions' cover (and are too slow to effectively ambush the same prey).
Cats were domesticated for entirely different reasons. They're a vermin-management service, eating mice, rats and small birds that would otherwise eat our stored grain. The process for this was less directed than with other domestication events as humans basically created environments that are great for small cats that eat mice and the cats moved in. I can't see how you'd start off with man-eating cats at this point, and breed them until they live off mice.
A possible solution
So, what you're looking for is a set of circumstances that would mean it's desirable for humans to work together with big cats, and importantly that it's desirable for big cats to hang around humans as well (rather than just eat them). Domestication (especially early domestication) is a two-way process. For wolves, proximity to humans (and ultimately participation in joint hunts) meant that they increased their success rate for getting food, and for cats it meant a steady supply of small vermin for them to eat concentrated in one area. You need a situation that's beneficial to both parties (and the more proximate that benefit the better).
You're going to have to invent some things to engineer this situation. Perhaps some sort of animal that needs the physical strength of lions to bring down, but perhaps has a level of armour plating that means it needs tool use to butcher (or maybe has poisons that need cooking to remove, but that is a longer shot as lions would have quickly learnt not to bother hunting those).
Or perhaps there are migratory animals that are susceptable to ambush-predation which gives lions a glut in part of the year, but for the rest of the time the game around is only really vulnerable to persistence predation. In the early days, humans scavenge off lion kills in the plentiful season (or drive them off them), and lions drive humans off their kills in the fallow season. Over time, this develops into humans leaving parts of their fallow-season kills on the periphery to keep the lions fed (and stop them driving them off their kills), and lions permitting humans to scavenge the bits of their plentiful-season kills they don't need.
The reason I put this option second is that the benefits are less proximate, which means it's harder for both parties to see the other as a benefit and not a competitor. It's also lacking a truly collaborative hunting aspect which would reinforce the thought that both animals are part of the same group. However, it's a bit of an easier option to extrapolate from so we'll use it for the next step.
So what might that mean?
In terms of evolution, I think you're right that they would reduce in size slightly and become more co-operative.
You might see them become faster and more greyhound or cheetah-like as that would make them better at ambush predation, and they wouldn't have the fallow-season enforcement of persistence predation checking their evolutionary change in that direction.
You could see breeds a bit like servals being bred for catching birds, or large guard-breeds that spend most of the time lazing around in one location but are powerfully built. Cats can also climb where dogs can't, so they might be breeds of use for hunting arboreal creatures (although, alongside the bird-catching variants, would need to be smaller and change their hunting behaviour so would probably be later developments).
Basically, in this circumstance (and any other circumstance), you need to trade the benefits of dogs (high endurance) for the benefits of cats (high acceleration) and see what uses humans could put that to.