Tokyo to Kyoto ... the long way?

Ok, I'm going to Japan in October and during the trip I'll be traveling from Tokyo to Kyoto. Now I was going to use the simple method of just catching the Shinkansen but the day I'll be traveling appears to coincide with the Japanese Grand Prix. This means that it is likely that the section of the journey from Tokyo to Nagoya could be very busy.

Hence I've been looking at alternative routes. I'm not in a hurry so I've been looking at taking a longer route and seeing some more of the country in the process. The route I am considering at the moment is as follows, (from Hyperdia):

Tokyo -> (Shinkansen Max Toki 315) -> Echigo-Yuzawa -> (LTD EXP Hakutaka 6) -> Kanazawa -> (LTD EXP Thunderbird 26) -> Kyoto

(This journey takes quite a while but goes through areas of Japan I haven't been to (yet) and hence could be interesting).

My question is regarding using the JR Pass for this, in particular for the portion of the journey between Echigo-Yuzawa and Kanazawa. Reading around I've been seeing conflicting information on whether this is over JR lines or not and hence whether it would be covered by the pass or not. How can I find out whether the JR Pass would cover some/all of the journey and if only some of it how much extra I am likely to need to pay (preferably before I go to Japan so that I can see if it's reasonable or if I need to work out another plan) ?