You can always call the Park headquarters. the State Police should have the info for you on how to reach them.
As long as you bike on the paved roads, you shouldn't have any bear problems, but you never know. It always pays to ask about bears when you enter the park.
If the park is open for free, take it to the bank that there will be people walking and biking, no matter how bad the weather is. This cold snap won't last long- I would hang until next week, if that's possible.
If not, and you have the rain gear, decide for yourself if the ride on a stormy day is something you might like or not.
This is a great time to see the park, but vigilance is needed around any and all wildlife. Mother chipmunks can be just as cranky if their babies are around as mother bears, or elk, or buffalo. Remember that they are all wild. If you give them the road, almost always, you won't have trouble.
Bear bells do work. If you don't have any, I think the Indian brass Hindu-Gypsy-Hippie bells are the ones most bears really love the most.