It is said that December is the time when Mount Fuji gets most beautiful. The top of the mountain will be covered by snow by then, and the percentage of getting sunny is very high in December. If you are a climber, hiking and climbing mountains during this time is recommended because it won't be too much snow in the mountains yet.
I don't know if they have English speaking guides, but there is a "Nordic Walking" program running throughout the year. A guide will walk with you around the lake => http://www.fujisan.ne.jp/fit/en/outdoor_en.html
Around Lake Kawaguchi might get pretty cold so don't forget to be prepared for winter mountains!