Free: Contests & Raffles.
If all you are needing is burger ground, I can do that here at the house. Grinder, mixer and vac pack. I am in Kent. If it is more than that, i dont know. Never even set eyes on a dead cougar so I dont know how it should be cut up.
I never killed one but the guys I know who have just take the back straps and most of the big muscle groups off of the hind quarters and leave the rest. Most processors usually have a minimum so unless its a monster you'd probably be paying a ton for not a lot of meat.
Stewart's Meats, on SR 507, between Yelm and McKenna. I cannot vouch for their service, but they do handle wild game.