My personal favorite onsen places are:
(My main interest is rotenburo and nice atmosphere. I donft know that much about water quality in order to be able to judge. )
http://bicycletraveljapan.blogspot.com/search/label/Onsen+Practicaliti...Kurokawa onsen: specially yama mizu ki, but I would say the attraction is to visit several. Should be within your budget.
Osorezan: bathing in ph 2 water in front of a temple in a sulphuric plain on top of a mountain. Staying in the monastery was also nice although I wasnft a big fan of their dinner. Should also be within budget. (You can also go as day guest). Youfd need to check if they are already open when you go.
http://bicycletraveljapan.blogspot.com/2017/10/day-13-hakodate-to-oma-...Hirayu onsen. I frankly only went to one onsen there (Hirayu no mori) as I had really bad weather when I was staying. But I guess for a real onsen lover you could stay in that valley for a week hopping from rotenburo to rotenburo.
Tamagawa onsen (Akita): there are 2. Shin Tamagawa and the old Tamagawa. I only went to Shin Tamagawa. The rotenburo isnft very attractive. But the inside baths are. Itfs like beating in a cathedral of onsen. Water at ph 1. I liked it, my husband found out a bit painful. The landscape around is otherworldly.
Moku moku onsen in the countryside between Koka and Iga Ueno. From the outside itfs a bit strange, but the rotenburo section is big and gets some landscape.
Ochiairo in Ito peninsula (ochiairo.co.jp). This might be above your price range but is an original ryokan from about 100 years ago. It also has a bookable private rotenburo.
If you donft mind mixed bathing, I tried these two and liked both of them:
- Kuronagi onsen in Kurobe gorge. Possible as day visitor or staying guest. Also has a small female rotenburo.
https://bicycletraveljapan.blogspot.com/2017/10/day-27-kurobe-gorge.ht...- Fukeno yu in Hachimantai (close to Tamagawa onsen). Bathing practically directly in a mud volcano. It also has a female only section and maybe also a male only section. I only went to the mixed one.
If you are also staying a few days in Tokyo, I like these two:
-Yumorinosato in Chofu ( there is also a nice temple close by with famous soba)
Yukaisoukai in Kurihira. Good dark water and very different temperatures
And if you are flying out of Narita, I can recommend the Narita view hotel. It has an onsen with 3 different rotenburos and onsen water. (One was closed now in November but is planned to reopen in December). The hotel itself is one of those large airport hotels, but for a last soak the morning before your flight, itfs ideal.
None of these per my knowledge have private onsen (unless otherwise stated) but then for me itfs about many different onsen, rather than having one for me alone. Most should be very well inside your price range.
Links to all these onsen are in the first link I gave above.
Enjoy the soak.