Alpine Route or Kamakura or Hakone

Hi, my friends and I are gonna visiting Japan on May. We will arrive at Tokyo and depart from Osaka.
Our fixed itinerary are :
day 2-3 : Tokyo (asakusa, ueno park, shinjuku, shibuya, disneysea),
day 4 : Gotemba Premium Outlet, Kawaguchiko (Kachi kachi ropeway), sleep at tokyo.
day 5 :
day 6 and 7 (flexible) : Kyoto (fushimi inari shrine, kiyomizudera, gion, etd),
Day 8 : Osaka (Dotonburi, Osaka castle)

Which one do you suggest for day 5 between Alpine Route, Kamakura or Hakone (for its lake ashi and la petit prince museum?) Heard that Alpine can be so expensive, but the snow wall only happen on April and May... Which one is the worthiest?
Can we go to Kawaguchiko and Gotemba and back to Tokyo for 1 day?

Which one do you suggest for day 5 between Alpine Route, Kamakura or Hakone (for its lake ashi and la petit prince museum?)

I suggest the one that you and your friends are personally most interested in (while considering your budget).

Indeed, the Alpine Route is by far the most expensive, but you could elegantly visit it on your way between Tokyo and Kyoto.

Can we go to Kawaguchiko and Gotemba and back to Tokyo for 1 day?

Yes, if you start early and travel at a relatively fast pace, it can be combined into a single day. It also depends on how much you want to see around Kawaguchiko.

the problem is we hardly decide where we want to go, love to see the snow (hate the price though), but curious on Lake Ashi and Kamakura... xD

For Hakone, if we going to kawaguchi and gotemba PO, and spend the night in Gora / Sengoku, do you know any ryokan with onsen there? the nice one but not expensive, as we travel on budget...

thank you sooo much!

There are lots of ryokan with onsen in these areas, but I am sorry I don't know any specific one that I could recommend.