Hakone Ryokan Onsen

Do you have to book at a ryokan in order to use their hot spring facilities? I am planning on going to Hakone for a day from Tokyo. The trip is mid may. Searching around, I found several traditional ryokan that look nice but staying at them is like $140 per room. I'm staying by myself. First off, is it worth it to take a bath at night under the stars? Or should I just go to a public bath? Are there any inns that allow the public to use their facilities in Hakone? Any recommendations as to where to stay for a traditional experience under $150? I'll have a JR pass so transportation costs aren't included in that $150. Here is where I'm thinking of staying

http://www.japaneseguesthouses.com/db/hakone/shunkoso.htm

You don't have to be a guest to use the onsen at many ryokans - usually the hours are limited for non-guests, though.
if you go to:
http://www.secret-japan.com/onsen/
and look for Tenzan Onsen in Hakone they have a description of the baths - it sounds pretty nice.
more info:
http://www.asahi-net.or.jp/~UE3T-CB/index_e/spa_e/tenzan_e/t...
If you do some google searches for hakone day onsen, you can find a wealth of information, some of it more helpful than others.