As you mentioned, you do not need to pay for the bus fare (1,750 yen) if you have the Fuji Hakone pass. It is all included in the pass.
The thing is that you cannot book your ticket for the bus from Shinjuku to Kawaguchiko or from Kawaguchiko to Shinjuku online in advance as you need to pay for the fare (1,750) online to complete your booking.
There are two ways to book your tickets as below.
1. Go to Shinjuku Expressway Bus Terminal or call the ticket office at 03-5376-2222 as soon as you arrive in Japan and make a reservation. You do not need to make a payment unless you book online.
2. Call the ticket office at +81-3-5376-2222 for reservation. English is available.
Another tip about booking this ticket is that it is easier to book a ticket from Kawaguchiko to Shinjuku, compared to booking a ticket from Shinjuku to Kawaguchiko. So, I would suggest you travel from Shinjuku to Hakone first, and then go to Kawaguchiko and come back to Shinjuku. Please note that it is difficult to book a ticket from Shinjuku to Kawaguchiko on the spot as the route is very popular for tourists.
Hope this is clear enough to make a plan for traveling Japan!
Have a nice trip in Japan!