Bicycles need a ticket provided they are not foldable or transported in a TranZbag.
You must not buy a Supersaver ticket for the bicycle itself - Supersaver tickets are valid for humans or dogs only.
You are welcome to travel on a Supersaver ticket, yet for your bike you either need a normal half fare ticket or the Bike Day Pass (if the half fare ticket exceeds the cost of a Bike Day Pass being CHF 14), plus a reservation for IC trains, where applicable.
Does my answer clear up the confusion?
By the way, I suggest downloading and using the SBB Mobile app. The ticket purchase is easier using the app, especially when travelling with bikes.
And, 1st of August is our national day, it is considered a Sunday, hence you will need a reservation for the bicycle space on Intercity (IC) trains. If you travel on the IR46, there is no reservation needed, bicycle transportation is, nevertheless, subject to available space.