The main difference between climbing Mt. Fuji in July/August vs. September/October, is the number of infrastructure and emergency services available on Mt. Fuji. Due to the lack of mountain huts open on Mt. Fuji after mid September, many sources online will portray Mt. Fuji as being closed to climbing after August. Although the climb is definitely more difficult to navigate without previous knowledge beginning September, the climb is still untechnical and doable September/October for novice climbers.
The temperatures do drop quite a bit beginning September/October so it is most certainly not wise to climb at night to reach the summit for the sunrise. This is just asking for trouble as there are no mountain huts open at the summit which during the season serve as emergency shelter purposes among other things. For this reason, during our off season tour, FMG will climb to station 7 on day 1 and make the summit attempt during the day after watching the sunrise from outside our mountain hut at 3,000 meters.