The idea of a cougar or other big cat predator is not a new one, I've seen it (and even put in my two cents once before) many times on the wish list and I think it merits some real thought by the developers. Below, I'll cover my personal in depth take on a cougar in TLD and how it could fit in with the current threat creatures.
To start off, cougars would be at least as rare as moose are, if not more rare, with only a handful spawning in any given game. There could only be one in any given region and similar to moose their passive behavior would be to patrol a small area around their den. This area could be marked similarly to how moose areas are, but with less obvious scratch markings on trees. Central to this area would be a den, a small and hard to notice mini-cave that would be its own enterable area. It would be small, smaller than any non-transition zone cave with the chance to spawn a frozen corpse, evidence of a cougar's previous meal.
The cougar would operate in one of four different states: Patrolling, Stalking, Hunting, and Fleeing
In its Patrolling mode, the cougar AI would have two 'sense' rings around it. Maybe 100 meters and 75 meters in diameter. If the player enters the first ring they might receive some sort of audio/visual cue (anything from the character mumbling something or a message similar to the thin ice warning) letting them know that there is a cougar nearby. There also might not be a cue for a more difficult cougar. In either case, if the player lingers in this first ring they will eventually be detected by the cougar and if they enter the second ring at any time they will automatically be detected by the cougar. At this point, the cougar goes into its stalking state.
In Stalking mode, the cougar has detected the player and will actively pursue them from a distance. The AI will try to position the cougar behind and a certain distance away from the player, and if the cougar is within the player's viewing area it will not move. There, again, might or might not be some sort of cue that the player is being stalked. Perhaps the world sound volume would decrease with the player's footstep volume increasing, something subtle that might not be very noticeable.
To end a cougar's stalking mode the player would need to either leave the region for a few days, aim at the cougar (in which case it would retreat beyond view distance and there would be a small random chance that it would revert back to patrolling mode), hit the cougar with a projectile weapon (in which it would go into fleeing mode), or drop a decoy which may revert the cougar back to patrolling mode.
While in stalking mode, the player is still very much safe from the cougar. However, if they get a certain number of afflictions/debuffs (anything from being overburdened to a sprained limb, to parasites) then a stalking cougar will go into hunting mode.
In Hunting mode the cougar will try to attack the player from behind, charging at them on quiet paws. Marine flares will keep the cougar from charging, but torches, fires, and road flares won't deter it. Distress Pistol rounds will cause it to flee and revert back to stalking mode, and a direct hit with a projectile weapon will cause it to flee or die (if a critical shot is made).
If the cougar's charge is successful, a two-stage QTE struggle ensues.
In the first stage, the cougar pins the player onto their chest and begins attacking from behind. The player needs to manipulate their movement keys to unpin themself and roll over. Once successful, the second stage is the same as any wolf struggle, with the player given a choice of tools to use to beat the cougar away. During the struggle, all clothing items take damage and debuffs such as bleeding, lacerations, sprains, and the like can be taken by the player. If the player manages to defeat the cougar, it will go into fleeing mode.
Cougars will not bleed out, though they will leave blood trails for a player to follow. Instead, in Fleeing mode, a cougar will return to its den for a certain number of days and will wait there to heal back up before reverting back to patrolling mode.
In order for a player to kill the cougar, they'll need to get an instant kill with their weapon of choice or pursue a fleeing cougar back to its den. After entering the den, the cougar will give a warning yowl before charging. If the player is fast enough, a successful hit will kill the cougar. Otherwise, they will have to engage in a final wolf-style struggle to kill the cougar.
Cougars will yield a good portion of guts and wolf-like lower calorie meat. The true prize will be their pelt, which can be crafted into two very useful items. The two items I thought up for cougar-coat crafting are the Cougar Cloak and the Cougar Bedroll. Both would require a single cougar pelt with some other items to craft.
Cougar Cloak: unlike the moose-hide cloak, which goes in the outer layer slots, the Cougar Cloak would go in the accessories slot with the wool ear wraps and moose-hide satchel. It would offer some warmth, wind resistance, armor, and water-proofness but be obviously heavier than the ear wraps and only work in the outer slot. Its main bonus would be a smell-resistance, where the first scent bar is blacked out and anything that would raise a player's scent bar by one would be effectively negated.
Cougar Bedroll: the cougar bedroll would be a middle-grade bedroll, with warmth and weight stats putting it between the bearskin and basic bedrolls. It wouldn't increase wolf fear as much as the bearskin version, but it would protect against animal attacks.