The lowest prices I can find in the area for those dates is the Whitetail Creek Resort in Lead. I'd contact them now before they book out, it will be starting to get busy then.
Note that it's pronounced leed, not like the metal.
There are several sites that allow homeowners to rent out their homes (including secluded 2-3 bedroom cabins in the woods). I've done a cursory search in the Rapid City area for your stated dates and found several in the $200/night range. Remember that's for entire houses on acreage! If that seems like what you're looking for, do a search on the terms 'VRBO' (vacation rental by owner), 'vacation rental', 'homeaway', etc. - also including 'Rapid City SD' in the search.
I hadn't even considered that they would be foolish enough to hold the rally this year. Traveling at that time is always a little extra challenging, and in a normal year, that would be something to keep in mind, but not necessarily keep you from doing the trip. However, anyone going to any event like the rally this year is someone I would assume is someone who isn't going to care one bit about your health or the health of anyone else around them. It's safe to assume that social distancing and masks will not be something you should expect to see.
I know it's not what you want to hear, but the Black Hills will still be there next year and the year after that. Traveling anywhere this year comes with challenges, but the Black Hills on the same week as the Sturgis Rally during the pandemic would be up there with flying to Florida to go bar hopping in terms of bad ideas. I really would recommend changing either your timeline or your destination.
Personally, I would cut out the Black Hills completely, considering what Michael said. You would be getting there too close to the opening of the rally, people will be getting there up to a week early and going exactly where you want to go. Last year, I traveled across SD on I-90 during the week before the rally, and there were a LOT of motorcycles and RV's out on the road all heading west.
The other option is to delay the trip till a week after the rally.
If that means spending more time in Cody, great. If that means doing as someone else suggested, looking at Glacier instead, great. If that means putting this whole trip on hold for a while, great.
Thanks for the help all. Updates have been made and everything is good to go. Good point on booking Yellowstone reservations early. I was lucky and they had one opening at Grant Campground that fit our schedule.