Firstly, if you’re going to Japan during the summer there’s a JR Pass Giveaway going on right now at http://www.onegreenbicycle.com/jr-pass-giveaway/
I think your itinerary works well. Toyama is more relaxed and I like the scenery better but it’s definitely a personal preference on how you want your trip to look. And hakone has a lot of great onsens in the area around Mt. Fuji. I think a fun option is to go to an onsen with a view of Mt. Fuji if you like onsens. If you’d rather get the perfect lake shot, then the lake would be your best bet.