Just wanted to try out a trip report so this is actually something i did early last October. First a little intro...
The 'official' open season for climbing Mount Fuji is just the months of July and August, however you can avoid the massive crowds by going late. The down sides are that all the huts and touristy stuff is closed, if you like that sort of thing, and of course the weather. Also, as the trails are supposedly closed then its possible that someone could object, but when I went there were a few other hikers around and no one seemed at all concerned - you just need to step around the barricades. Doing the popular overnight hike, i.e. summit at sunrise, would be harder out of season due to the lack of operating huts, but I suppose you could always kip in the car till the start time. This is another advantage of the off season as in peak season the highest car parks are closed to the public to reduce congestion and you have to take a shuttle bus from lower car parks, assuming you are driving in. In season you can take a direct bus from Tokyo, but I think only tour buses operate outside of that i.e. you just get to see the trailhead.
Regarding the weather, summit temperatures on Fujisan are around 20C to 25C lower than nearby Tokyo, so you need to dress appropriately whatever the season. On the days preceding my hike the summit temperature at the peak weather station was around -15C, but on the day it was a more typical -4.5C. Despite being colder, I would pick October over September due to the lesser chance of Typhoons. There is little to no shelter on Fuji, and wind is potentially the biggest danger (assuming you are dressed properly).