When I was doing research in YNP, bears weren't seen on the main roads, so, carrying stuff on a bike shouldn't be a problem unless they are now as common as bison along the side of the road. At the time, the most bear activity was on Mt Washington. It might have changed since then, with wolves and elk dispersed differently because of the wolves. Ask any ranger and they will tell you were the bear activity usually is located.

As for campsites, campgrounds have central locations to wash dishes. Just don't cook in your tent, leave all food and personal hygiene things (including towels that might have soap in them) outside the tent, and the clothes that you cook in-- also leave them outside the tent. Dried food is a good suggestion.

There are also bear-proof food containers you can carry on your bike. (I think) it was miles2go who featured one in a photo in his recent bike trip in Alaska; ask him about it. I saw one in REI, but, have no experience with them.

Enjoy the trip. YNP is a special place.