Mt Fuji to kobe

Hi.. We are a family of 4 travelling to Japan in August. Want to do the overnight climb at mt. Fuji and upon descent, move to our next destination, Kyoto/ Kobe. What is the best and quickest way to get from Fuji 5th station to Kyoto or Kobe (whichever is easier). I understand there will be some transfers but do not want to go back to Tokyo (as some websites suggest or is that really the simplest way?)