Dogs versus large cats.. as others have mentioned, if there are enough dogs, they might be able to distract the cat long enough until you can shoot it. Also those Rhodesian Ridgeback dogs (African Lion Dogs) were not bred to actually kill lions, but to help hunting them and in a group of a couple of dogs corner the lion so the hunter gets a better shot at it.
I have been in Namibia last year where they still are alowed to hunt the African Leopard. I spoke with the owner of a hunting farm there and he told me that first of all (eventhough they have a population of about 20-21,000 leopards in Namibia, there is never an incident between humans and leopards.. except when they are being hunted.) So if they shoot a leopard and only injure it they have to go after it to kill it as a wounded leopard is not something you want to have around on your farm.. Problem is to stalk the wounded leopard as it usually disappers in thick bush.. if you follow it, it will circle you and attack the whole hunter group from the side or back. Within a couple of seconds a wounded leopard will injure or kill a hunter group of 3 or more men.. Just for those who think you are ok with your .357 in your backbag.. A group of hunters with large caliber rifles and being fully aware that there is a leopard close by and having their rifles ready to shoot.. they can be beaten up by a single leopard before anyone can even fire a shot! Happend more than once!
So what they use to stalk the wounded leopard is dogs. Usually some aggressive small terriers. These dogs are small and don't have any fear. They are somehow bred that way.. Such a dog, even alone, will attack a leopard or large snake or whatever is moving.. not that the dog will survive.. it just goes for it and often dies in the attack. So they use around 5+ of these dogs to stalk the wounded leopard.. they find the leopard and attack it so the hunters will know where it is and can go and shoot it.. And yes, he told me that usually some dogs will die in such an event..
Just mentioning this as I guess the African leopard and some of these US cougars seem pretty close in size..
On the other hand.. blade versus large cat.. well I would never want to experience that and would rather have a gun on me but the Maasai tribe in East Africa proves that it surely is possible. For a young man to become an adult he has to go out and kill a lion! Not a leopard or a hyiena.. no, a full grown African Lion!! This has been practised for hundreds of years and eventhough rather uncommon today (as it is also against the law) it still happens today. Mostly when a lion is killing their livestock it is time for a young man to become an elder while going out and killing the lion.. And they don't use some high end blades in INFI steel or such.. I have handled such Masaii spears and that's not something very robust to 'fight' with. A great stabbing weapon that has to be aimed very precisely.. But honestly if a lion would come carging at me I most likely would die of a heart attack even before he reached me.. even if I would hold a large gun in .375H&H or .460WM or whatever.. So I have the biggest respect of these guys going after lions with nothing but some thin spears and a leather shield..
So.. long story short.. blade versus large cat.. doable.. if you are a Maasai and have been spending all your days since childhood learning how to do it.