First thing I do with the Crystal II inlet socket is remove its filter pack, chuck it away, and refit the empty canister. Those filters are meant to be replaced every month of use, (fat chance of that at over £7 a pop) - Porton Down store fewer bio-hazards than those things.
Yes, there is a non-return valve in the socket, so there shouldn't be any others in line.
Remove the pipe from the back of the socket, open both taps, and try to blow through it - it shouldn't be too difficult to force water (or air) out of the taps.
But, from what you suspect, those extra NRVs need removing anyway.
Another thing worth checking, and often overlooked, are the tap's spout filters.