Nothing wrong with that idea, it's just impractible to mix an interrail trip with a bike trip. Renting bikes can get expensive, too, but I doubt this would exceed the amount spent on buying bikes/transport fees.
BTW, who made this itinerary? Everyone of you got one week to scheme? Nevertheless, you should take a glance at your itinerary again and do some more research:
a) you claim you intend to rely on nighttrains a lot, but note that there are no nighttrains Paris-Bern, Lugano-Bologna, Ljubljana-Cesky Krumlov or any other stretch of your journey with the exception of Prague-Brussels (yet there's not even a direct train, you'll have to change at least once).
b) as far as I know, the ferry Ancona-Zadar is not included in interrail, you won't even get a reduction.
c) you're spending a larger chunk of your time in Croatia/Slovenia, yet your pass will likely not be very useful there, as trains won't get you to/from Novalja, the few trainrides you'll do in Slovenia are rather short and train travel in this part of Europe is still rather cheap.
So a '10 days in 22 days' interrail pass would be more appropriate.