In addition to the extra drive time on Day 10, I was also thinking of the extra drive time on Day 9 since the OP wants to drive the Beartooth that day.

It's just a little more windshield time. When we visited Yellowstone we had a 550 mile detour just to get into the park. Relax, you'll be on vacation.

Old Faithful Lodge & Canyon Lodge aren't so far apart from each other to get upset about; you could actually stay in either one for the entire trip & not have a problem. The park route is roughly in a "figure 8" shape, with OF Lodge being in the "bottom" section of the "8", and Canyon Lodge more at the center of both loops. You could drive the bottom half of the "8" one day; do the top the next, saving time to do activities within each particular loop. Best advice is to start your day early (best time to see animals/get jump on park traffic).
You could leave either Yellowstone lodging (skip Gardiner lodging), spend time in park 'till noon/drive Beartooth Hwy to Billings/stay for the night (further down road than Red Lodge for morning exit).
Leave Billings AM, get on I-90 to Hardin, MT to visit Little Bighorn/Custer Battlefield. Continue on I-90 to SD. Battlefield/Museum can be seen in 2-3 hrs; less if needed. Devils Tower could also be done as a daytrip from SD, if you wish - we left Custer,SD after breakfast, drove there & back by 6:00 PM. Interesting to see for a short time, but not much to do there other than watch climbers scaling it. If you have the time to stop on your way TO SD, that would be a better use of time. Personally, I would choose activities in Black Hills over Devils Tower for a first trip with sons.