I haven't kept up with Mt. Fuji's conditions this year, so I'm not sure if snow has already started falling at the top yet, but if the snow isn't bad there's no reason you shouldn't be able to climb it at this time of year. This year's Autumn weather has been exceptionally calm. Usually this is the middle of typhoon season, but so far we've only gotten two typhoons and the more recent one didn't even hit land. So I'm guessing Fuji's pretty good for a climb.
As you say, definitely don't underestimate the mountain. It's freezing up there in the summer, and by this time of year temperatures are definitely winter-like. Also the winds around Fuji can be very strong and storms very sudden, because Fuji stands alone in the terrain and creates its own weather. Just make sure you have good rain gear, warm clothing, and some kind of emergency shelter. I'm not sure if the huts are still open (I suspect they are not), so don't count on finding refuge or food there. Carry your own water, because there are no water sources. I would say check the bus schedule for getting up to the 8th Station, where you would start the climb. It may be that the season has ended and no more buses are running, but I'm not sure. You could walk all the way up the mountain from one of the bus stops that run along the base of the mountain (and are open all year long), but it would take you twice as long, probably about two days just to get to the top, so a three or four-day trip, instead of the usual two-day trip. If you have experience climbing 3,000 meter (10,000 ft, but actually almost 4,000 meters) peaks then this should be an easy climb: just go zig-zagging up the trail.
I'd say go from the Fuji City or Kawaguchi side, because they are shorter climbs, and then descend to Gotemba, because it is all volcanic ash, so easier on the legs, though very long. It's also easier to head back east to Tokyo or Yokosuka from there. You may want to wear shoes that you don't mind trashing, volcanic ash and pumice can be hard on your shoes. Also wear some kind of gaiters or protection for your ankles because the ash gets into everything. If you climb during a clear day bring a hat; it is hot and the UV rays strong. If you climb at night, make sure you have a good light. And finally try to get to the summit just at or a little after sunrise. Before the sun's rays hit the top it is bitterly cold up there and the wind is usually stronger than along the flanks.
I hope I didn't leave anything out.
Good luck and have a blast!
Edit: I used to live right at the foot of Mt. Fuji and saw it every day of the year for five years. I've climbed it only once, but from my views of the mountain the weather is usually pretty good at this time of year and the snow usually still just a dusting. I don't think snow will be the problem, at least not until about the middle of November, but wind and bitter cold definitely would be. I know people who've climbed it in winter, so it is climbable all year, but they were seasoned mountaineers who regularly go to the Himalaya. Please be VERY careful when planning this trip and turn back if it looks at all like it might be beyond your abilities. I met a Canadidan right near Mt. Fuji once who attempted to climb Fuji in winter wearing nothing but a Canadian postal worker's uniform and a track suit jacket. I had to ask for local help with lending him a down jacket just to make sure he didn't perish up there. I don't know what happened to him, but his last words as he waved from the bus were, "Don't worry! I'm from Quebec! I KNOW what cold is!"