Nagoya - Takayama - Nagano - Tokyo

Hi, we are a family of 3 including a 5 year old who travel for free.

Am wondering if we should be getting JR Hokuriku Arch Pass for the following:

Day 1 - Hida Limited Express from Nagoya to Takayama
Day 3 - Takayama to Nagano
Day 5 - Nagano to Hakuba or Karuizawa
Day 6 - Nagano to Tokyo

For Day 2 & 4, these will be local buses as I believe the pass do not cover them
Day 2- day trip to Shirakawago
Day 4 - day trip to snow monkey park

We are sticking to 1 activity per day as we have a child to cater to and not sure if the pass will be worthwhile and indeed cover all the places above. We did consider driving but the difficulty to secure snow tyre and the one way return fees do add up.

We are also concern about the traffic and worry that the snowfall and peak season will means long jam and missing out on fun as we are on the road, If I'm mistaken and the road will not be as conjested, we will be open to rent a car from Takayama on Day 2 and return at Nagano on Day 5.