Eurail themselves don't seem to want prospective travellers to be able to check pass holder seat availability.
Fortunately, you can check Eurostar seat availability here (€30 in 2nd or €38 in 1st class):
https://www.b-europe.com/EN/Booking/Pass#TravelWish
I still see lots of available seats on that day but don't wait too long to book.
Brussels-Stuttgart and Stuttgart-Bern have optional reservations and can be booked via DB (use “Seat only”) (€4.50 in 2nd or €5.90 in 1st class, price per journey including all bookable trains).
Bern-Chamonix doesn't need any reservations. You can book some domestic Swiss trains but that's a waste of money.
Chamonix to Paris needs a TGV reservation from Bellegarde to Paris, which can be booked here (10 or 20€, depending on availability):
https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish
You have to do the math yourself to decide whether a pass is worth the money. You can use e.g. www.thetrainline.com to check normal (advance) fares.
Just a small point, as you are from Australia it is an Eurail pass that you would need.