A) pickup at our hotel (Shinagawa area)
B) 6 or 7am start.
C) Maybe 5-6 hours tops.

Taxi at 6AM to Shinagawa Station (or Shinagawa hotel shuttle or walk)
6:28 AM Shinagawa —Odawara 6:54 AM
¥3,830 on the shinkansen (26 min)
Taxi from Odawara to Hakone Ropeway Sounzan Station 30-40 min drive
Taxi from Odawara to Hakone Sightseeing Cruise / Lake Ashi 26-35 min drive
Taxi distance would be 20 km
Assuming taxi could fit four pax

Leave hotel at 6AM and be at a spot that offers great Fujisan views by 7:30 or 7:45
Save some yen and take JR regular train 80 min back to Shinagawa.

For Fuji Five Lakes that would be:
Train/subway to Shinjuku
JR Fuji Excursion to Kawaguchiko
Bus ride up to the ropeway base station
Wait
Ropeway up for great view
(that is all a bit of a madhouse experience)