Hakone and Mt Fuji from Tokyo

In Haneda: 27 Mar
Out Haneda: 7 Apr
27Mar to 5Apr - Tokyo (incl day trips)
5Apr to 7Apr - Hakone 2 nights

Question 1
How should I fit in a trip to Mt Fuji and Lake Kawaguchiko without staying there?

Question 2
Have shortlisted these 2 places to stay in Hakone. Have listed the link which shows access to the hotel.

Which is the choice if I'm looking for easy access (2 adults and a 3yo child with luggage) from Shinjuku train station?

On 7 Apr, will visit Gotemba Premium Outlets after checking out. Then will return to hotel to retrieve luggage and depart for Haneda airport.

Choice 1: Shinanoki Ichinoyu (Annex)
http://english.ichinoyu.co.jp/shinanoki/access/index.html

Choice 2: Laforet Hakone Gora Yunosumika
Get off at NakaGoura Station (on Cable car) to your right. 5 min walk from there

Appreciate your kind input!

Anyone can input their valuable advice please? Not sure if I shd do a round trip Mt Fuji and Fuji Five Lakes from Shinjuku Tokyo or visit the place while I'm putting up at Hakone. Will there be enough time?

Question 1
How should I fit in a trip to Mt Fuji and Lake Kawaguchiko without staying there?

for my opinion, u should consider stay 1 night at hakone-1 night at kawaguchi. but if not , that's mean u must go to kawaguchi then back to hakone using bus (covered by fuji hakone pass), but it more consumed time (about 1,5-2 hours 1 way from kawaguchi to hakone)

Question 2
Have shortlisted these 2 places to stay in Hakone. Have listed the link which shows access to the hotel.

i had no idea about the hotel