Itinerary Mount Fuji

I am making much progress planning our itinerary for our May trip to Japan and I am currently planning the last few legs of the journey :)

After exploring the Kansai area for about a week and before returning to Tokyo (incl side trip to Nikko), we would like to see Mount Fuji.

This is part of our itinerary and I would be glad for comments, feedback and suggestion:

Day 11 - wake up at Mount Koya spending the night in a temple, go back to Osaka, head to Hakone in late afternoon/early evening and spend the night
Day 12 - Hakone Loop, spend night on Hakone (or is it feasible to dl the loop and head to Fujikawaguchiko afterwards?)
Day 13 - make our way to Fujikawaguchiko, spend the day (possibly visiting Chureito Pagoda) and stay overnight
Day 14 - head to Tokyo

Does the plan make sense? Any recommendation to improve, what to see and where to sleep? Hakone and Fujikawaguchiko seem quite pricey... which pass is the best as I believe the JR pass only covers part from Osaka to Odawara if I am not mistaken.

Thank you so much in advance!

I think it is good. I would do the night in Kawaguchiko so your aren't rushing so much. But either way would work.