Shinjuku to Mt Fuji 5th Station

Hi, I'm 60 and have a heart condition, so I do not intend to climb up Mt Fuji. I just thought of being able to step foot on this famous mountain and take some photos. As I'm booked in a Shinjuku hotel for 23rd - 29th May, I can spend two days going to Mt Fuji and Hakone but prefer to return to Shinjuku each night. I am now deciding among 4 options:
1. Just join a one-day tour to both Mt Fuji 5th Station (think it's Gotemba?) and Hakone/Lake Ashi.
2. Just join a two-day tour, if available, and spend a more relaxing time enjoying these two areas.
3. Take a personal Hakone Free Pass to do a Hakone Round Course and view Mt Fuji from there within one day.
4. Spend 2 separate days going to Hakone on a weekday and then to Mt Fuji 5th Station on a Saturday as I read that there are only two direct buses from Shinjuku to Kawaguchiko 5th station on weekends in May. Is it cold up at Kawaguchiko 5th Station during this period, and besides enjoying the view, what can I do there while waiting for the return bus?
Finally, is there a direct way from Shinjuku to Gotemba 5th Station which I can do in one day? and what is the cost?
Thanks!

but prefer to return to Shinjuku each night.

I assume you have an good justification for such an inefficient way of travel.

I feel that the Sunrise 1-day bus tour might be your best option:
http://www.japanican.com/special/mount_fuji_tours/index.aspx?aff=exj

Finally, is there a direct way from Shinjuku to Gotemba 5th Station which I can do in one day?

Only by rental car, but I don't recommend driving in Tokyo:
http://www.japan-guide.com/e/e2024.html

Gotemba 5th Station is not served by public transportation outside of the climbing season (July and August).