Unfortunately, the Fuji Hakone Pass which includes the expressway bus ticket from/to Shinjuku is valid for only 3 day from the date of first use. So, the Fuji Hakone Pass might not be the right choice for you.
If you want to go to Kawaguchiko and spend a few days in Mt. Fuji area, I would recommend that you just purchase a round trip ticket from Shnjuku to Kawaguchiko from the following website.
Then, you can just take local buses which connect major sightseeing spots. No reservation is needed for those local buses.
Please see the link below for more information.
Other option is to use Q-pack tickets.
The Q pack ticket includes a round trip ticket from Shinjuku to Fujikyu Highland and a one day free pass for the Fuji Q Highland which is one of the most thrilling amusement park in Japan. As your children are over 12 years old, Fuji Q highland may be a fun place to enjoy all day long. There are also Fujiyama onsen (hot spring) in the same place, so you can enjoy hot springs as well.
You can take a local bus from Fujikyu Highland to Kawaguchiko.
As for the luggage, you can bring your luggage into the highway buses. Also, once you arrive at Kawaguchiko, there are a lot of lockers and places where you can leave your luggage.
I hope this information help you.
Have a nice trip!
Let us know about your trip!