The explosion from firing might be quieter but the rounds would still be noisy as hell unless they are fired at puny subsonic speeds. You can't get around physics that easily, unfortunately...Also, the limits of compressed air have been thoroughly explored already and they just don't come anywhere close to the power of explosive propellant or everyone would be using them instead as it would allow for gigantic magazine capacity and the end user could carry an vast ammunition supply for the same weight.
My advice is to keep this design but tell people it's a chem/ rail hybrid, it works perfectly for that, silent rifle? Not so much. The compressed air bottle could be coolant to mitigate the huge heat buildup associated with linear motor weapons and the rounds having a casing actually makes sense (it makes zero sense when it is an airgun).