Fishing Bridge is a prime area for grizzlies. Years ago there were many instances between bears and campers. However, since the park rangers have instituted strict guidelines for coolers not being left outside or grills not being left outside, the problem is very minor nowadays. You will be given directions on how to keep a clean campsite. It will be no problem for you.
We've cooked outdoors all summer on our Alaskan trip in grizzly country and never had an issue - along with many, many other campers. It's simple.... keep a clean campsite and you won't be bothered.
We've also volunteered in national and state parks and I can tell you firsthand experiences with messy campers or those that think the rules aren't meant for them and bear activity around their site and eventually they were evicted from the park. The rangers will take no excuses for a dirty campsite.
Bottom line.... don't worry about it if you stick by the rules. Enjoy!!!