I live and work in Belltown, so I've got a good grasp on the lay of the land. I've got three distinct suggestions for you;
Buckley's: on 2nd and Battery is the best all around sports bar in the hood, it is however packed nuts to butts in there during all Seahawks games, mostly due to it being a de facto home bar for displaced fans of other NFL teams. Fun atmosphere, just too many people for my own personal taste.
Spitfire: on 4th and Bell another great all around somewhat upscale feeling sports bar with good food and lots of TV's. Gets good and packed and rowdy for Hawks games.
Belltown Pub: my personal favorite, mostly due to proximity to my apartment I suppose. Good classic pub grub, lots of TV'S, not quite as packed as the other spots usually but just as rowdy and a good all around atmosphere. I'll most likely be there in my Wolf Gray Richard Sherman Superb Owl jersey starting the SEA-HAWKS chant. Giant dude with long red hair and a burly red beard lookin like an extra straight out of Vikings lol.
Bonus venue, The Whisky Bar: 2nd and Blanchard. Not really a sports bar per say, but it is my favorite all around watering hole. A cool British style pub owned by fun British expats who, along with their undying devotion to EPL soccer, have recently become smitten with American football and in particular the toughness, tenacity, and all around philosophy of badassery that is the Seattle Seahawks. It can be a fun place to watch the game, but not always very lively. A great place for a post-victory beverage, especially if you like Whisk(e)y, they have an amazing selection as well as fantastic craft cocktails. A couple of my buddies are bartenders there, and they know what they're doing.
Anyway, I hope this is useful information. Enjoy your stay in my 'Hood and maybe I'll see you out there. GO HAWKS!!