I live in Wyoming and have biked the park for 30 years.
From before the roads are opened to cars until closing date in the fall.

Yellowstone is doable - but there is considerable traffic and shoulders vary from moderate to none.
It really is best to cycle in the early, early morning - almost no traffic and lovely. Also into the evening.
You can bike late because all the campgrounds but one (PebbleCreek) have reserved hiker/biker campsites.

Yellowstone's road system look like a "Figure 8" with 5 entrances.
The lower, larger loop generally has good shoulders with the exception of Canyon-Lake.
The smaller, upper box has narrower roads with few shoulders.
(The Mammoth-Norris segment is under reconstruction and will be much improved.)

The West and East Entrance roads are excellent with shoulders.
The Northeast Entrance road is narrow but with much less traffic.
The North Entrance is steep, narrow, with hairpin curves.
The South Entrance is fairly steep with no shoulders - challenging northbound.
I always try to ride it late when there is shade and lighter traffic - camping at Lewis Lake.

I suggest a south to north route -
South Entrance, West Thumb, Lake, Canyon, Norris, Mammoth, North Entrance
From West Thumb to Lake you follow the lake shore - Gull Point Road is a nice back road loop.
Lake to Canyon is a stunning ride along the Yellowstone River thru the American Serengeti.
Canyon has great camping, services, showers, laundry - and fabulous hiking along the canyon rim.
Norris Geyser Basin is far less touristy than Old Faithful - esp. in the early morning or evening.

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US 89 is a sweet route up to Great Falls.
Use back roads on the east side of the Yellowstone River from the park to Livingston.

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If mid-summer, consider riding via Stanley, Idaho thru the Sawtooth Mountains.
Stunning scenery and cool temps rather than baking in the Snake River Plain.