So glad you asked! I love Yellowstone and go for 2 weeks or so every year.
When exactly are you going? What airports in/out of and approximate time? (So I can make suggestions for what to see on the way!)
I'd divide your stay with 2 nights in 2 places. I like to divide between Mammoth and Old Faithful but Mammoth and Lake works too. Do be aware that the Lake Hotel dining room will close Sept 2 this year for a complete kitchen reno.
Roosevelt is my favorite but closes on Labor Day every year. To me Mammoth or even Gardiner (gateway town) are good subs and it's still easy to get out to Lamar Valley for wildlife watching. (I see you realized that while I was typing, lol!!)
Here is a link with when things close in the Fall including dining options.
https://www.nps.gov/yell/planyourvisit/operating-dates.htm
With 4 nights in Yellowstone, I would do 3 in Mammoth and 2 at OF OR 2 Mammoth, 1 Lake or Canyon and 2 Old Faithful. I am personally not one for switching lodging every night but it doesn't bother others. It's your trip!
Do be sure you book your lodging with the official concessioner, Xanterra, at www.yellowstonenationalparklodges.com Do NOT book thru a 3rd party provider with names that "sound official" they are not connected officially with the parks at all. They charge a 10% booking fee and have terrible cancellation policies.
I generally suggest people pick up a cooler at their arrival town and stock with breakfast and lunch items on the way to the park. If you are later in the fall the dining options will start to close. I prefer to have food with me in case I'm watching an interesting wildlife opportunity and don't want to leave to fill my stomach!
editing to add: Coming in over Beartooth will really depend on your dates. Tell me where from so I can also suggest an alternative if you hit wintry weather.