If I have a two-body system (particles A and B) which I measure the total momentum of, can I measure the position of particle A very accurately and the momentum of particle B very accureately, and then use the momentum of particle B and the total (conserved) momentum to give me the momentum of particle A with uncertainty satisfying $\Delta x \Delta p < \hbar/2$, violating the uncertainty principle?
A similar trick can be played with any conserved quantity and it's conjugate variable. What step or assumption is incorrect? I think that maybe the difference between measuring the whole system and measuring constituent subsystems is maybe not as trivial as I have assumed.
Following on from @AaronStevens comment, I suppose the question may boil down to 'Does the wavefunction of a particle change if we obtain new information about it without measuring it directly? If not, why not?'.
I think a similar question has been asked here (Violation of the uncertainty principle) but it wasn't quite formulated properly. This is perhaps what that author meant by their question?