The funny thing about wildlife habitat in the western Unted States? It's growing into the urban areas as certain species become very good at thriving among the humans and their vast supplies of free food and nice places to hide.
Funny you should mention this. Just got done watching a story on GMA this morning about a bear that took a dip in a residential swimming pool in Monrovia, CA.
Last week I saw two different stories about bears. One was captured in Steamboat Springs. The other in a heavily populated area Bucks County, PA, not that far from Philadelphia.
Saw one bear on the TA at the campground Jenny Lake in the Teetons. (Despite being warned that a bear had been seen around camp during the previous few days, an uncaring cyclist left his trash on his picnic table while he wandered off to try to get a cell phone signal.) It was a young one. Likely something had happened to the mother or she drove him off after becoming pregnant again early. A few pots and pans banged together scared him off.
As noted, your chances of even seeing a bear are generally remote, except maybe in Yellowstone where you won't be doing any wilderness camping unless you leave the bike behind and go into the backcountry. The three black bears seen during my rides on the Northern Tier wanted nothing to do with humans. They simply crossed the road and went about their business.
BTW...Unless you are going to be traveling up some unpaved road for a few miles, those U.S.F.S. campgrounds are not likely to be that remote. Many are right along the highway and can be heavily occupied depending on the location and time of year. Madison and Colter Bay campgrounds in Yellowstone N.P. were insanely crowded near the end of June. I did see some old evidence of bear activity (ripped up tree stumps) at the primitive ACA/U.S.F.S. campground on the east slope of Togwotee, but even that place is close to the highway. And with it's stunning view of the Teetons, it's worth any risk you might be exposing yourself to. Just use the bear box.