I'm not sure about a private onsen MD. But there are so many establishments with onsen it might be possible - just ask around when you get here. Also, when we've (family of 4) travelled to other parts of Japan I've found on occasion that a hotelier will say "It's very quiet at x time" and will let us look the door for a little while. If granted such opportunity please don't be tempted to, let's say, 'defile' the place with your partner.

They take a very dim view of such things.
There are a few places in Akakura Village and you'll probably find something you'll like. I don't like to make too many recommendations as some parties may be offended if not included (got to maintain the
wa). It's not a raging Niseko place by any means. Much more to kick back with a few drinks.
Niigata is famous for it's
sake and there are a few great local brands. Sechubai is the most famous in Japan - almost the Grange Hermitage of
sake. In the gift shop opposite the station you can buy some great Myoko-san
sake in a really cool earthenware bottle for about Y2,000.
Good luck! MG
P.S. Please come back on here after the trip and let us know how everything went.