There are NO back roads in Yellowstone N.P. - other than a few short segments of the old Grand Loop Road.
The longest stretch is Blacktail Plateau Road - one-way for cars - both for bikes, but care needed for opposite riding.
The National Park Service has a dual mission:
To provide for the public enjoyment AND to protect unimpaired for future generations.
As a long-time Wyoming resident I have seen massive tourism pressures on Yellowstone in all seasons.
4 million visitors per year is a lot for a remote, wilderness region.
Last year you had multiple serious incidents which resulted in damage to fragile environments -
As well as the deaths of tourists and wildlife. There is a reason for the regulations.
I support the current park policies which may restrict bicycle use more than I would like.
I have ridden every possible legal mile in the park - including when roads are closed to cars.
And, yes, having a cloverleaf interchange at Old Faithful is crazy.
But with careful planning, you can enjoy the park by bicycle - still.
All campgrounds except one have hiker/biker campsites.
Riding as early as possible (#1) and late can mean fairly empty roads -
Plus morning and evening are the best times to view wildlife.
Old Faithful is a zoo - Firehole Lake Drive has almost no traffic and Fountain Geyser is lonely.
A million cars are at the Artist Point parking lot, but the trail is empty 100 yards past the viewpoint.
And the Lamar Valley is wonderful with excellent hiking opportunities.
If you approach the trip with what the park can offer and how best to do that -
Rather than what you want and how to do that - then you can have a great trip.
PS - The are a few off-road opportunities immediately outside the park in national forest;
however, much of the adjoining lands are designated wilderness areas where bicycles are prohibited.
Not to mention that it is critical grizzly habitat.
There is a stunning route along thecrest of the Gravelly Range on southern Montana. July earliest.
Pic - Soda Butte Creek