I have to admit that I don’t know any of these onsen, but a search in Japanese brought up this page:https://journal4.net/sapporo-kazokuburo/
Scroll down and you will see a picture of each place where you can rent a “family onsen=kazoku buro “. It lists the price either per person or per time and has a link to their webpages.
They are all close to Sapporo. Some however only reachable by car (or at least that’s the short description in the access アクセス section. If you see car 車 written there you’d most likely need a car. If you see ... station ... 駅 then there is a train station somewhere close by. (Generally speaking. Unless it is 道の駅 in which case a rest station along a road is referenced).
Some of these places wil have inside onsen only. Others have outside onsen “rotenburo 露天風呂“.
Here an other page in Japanese https://www.recruit-hokkaido-jalan.jp/s/guide/sp/
Note that most of the pictures would seem from their public onsen, not the private ones!!! Eg the first one on that list: https://www.recruit-hokkaido-jalan.jp/s/guide/h00457
it is MAINLY a mixed onsen, but they do have a family onsen for rent. I couldn’t find out details but as it’s not written anywhere that it IS rotenburo I’d suspect it’s not.
Searching in English also this webpage came up. It seems to be all Hokkaido not Sapporo region only: https://selected-ryokan.com/?ryokan-cat=private-use-hot-spring-baths&location=hokkaido-region some however might rent their private hot springs only to staying guests.
Reservation will be key and you might need help for that from your hotel. I’d suggest you pick a few and discuss them with the clerk at your first hotel in Japan and get their input and ask them to make a reservation for you.
Onsen experience is one of the nicest experiences in Japan. I am already looking forward to my next soak in about 2 weeks time. Incidentally I’ll also go to Sapporo for the snow festival this year and will look for nice soaks as well. I have however never tried these private onsen. But I get it that for couples on a trip bathing together is a great experience. I had that only in ryokans and once in a mixed rotenburo in Hachimantai area. Which was great. But I get it that for beginners or shy people in general mixed naked bathing might not sound attractive.
Btw, in the female only section there is really nothing to be shy about. In Japanese culture bathing together is so normal that you see all kind of bodies. From very thin to overweight, from baby to old granny who needs help walking around. But no one is really looking. Everyone is relaxing together. (It’s the same on the male side I figure) The only thing to be aware is that small boys and girls are allowed on both sides. As a women I don’t think this is odd as as a mother or aunt or kindergarten teacher ... you’d be used seeing naked small boys and girls. But maybe for men it can be a bit strange to share the tub with a naked 5 year old.
Enjoy your trip to Hokkaido and the soak!