If you're driving a truck and have room for the bikes or are pulling a trailer, then you should definitely bring them, especially if you're going to be around for 2 weeks. Obviously inside of Yellowstone there are no riding areas, but if you want to spend a few days or the portion of a few days driving 10 or 20 miles outside of town, you will be able to ride.
I know that if I were coming all the way from Michigan and was spending two weeks of vacation in the area, it would be torture without my bike :-)
A couple of years ago my inlaws took all of us to Carlsbad/Oceanside area of California to stay in a condo on the beach for a week. I tell you what....after one day I had enough of the beach and was just itching for a ride in the Southern California desert/hills. Don't make the same mistake :p
Regarding the "bear problems" in Yellowstone, I wouldn't worry about it unless you're staying directly inside of Yellowstone and are camping. Even then, the majority of "problems" occur in the backcountry rather than the more prevalent campground areas (but that doesn't mean it can't happen). If you're staying in West Yellowstone then I wouldn't worry about it. The funny thing, in 25 years of going up to that area, the only bear I have ever seen in the wild was on the front porch of our cabin one morning (in Island Park), and not in Yellowstone.
There isn't a whole lot of info on motorized trails and opportunities in Island Park, but there are a few:
http://www.utahatvtrails.com/ATV.html (this is a great site a guy in Utah has created, but he does have a description of one of the rides I was referring to in Island Park....click on Trails on the left navigation bar and then scroll down til you find the Island Park Idaho link)
http://www.islandparknews.com (website for local newspaper)
http://www.islandparkidaho.com/
http://www.westyellowstonenet.com/island_park/community.htm
(Obviously within those sites, there are numerous other links as well).
Have fun.
Mike