Faradays law and Motional EMF contradiction

Sorry I misread the question.

The induced motional emf across the ring will be $B2rv$ as you have stated with the "top" part of the ring and the bottom part of the ring contributing the same magnitude.

The rails together with the termination $PQ$ define another loop with only the loop moving and the rails and termination stationary and the magnetic flux through that loop is changing.

The moving loop by itself itself does not contribute as the magnetic flux through it does not change and so there is no "extra effect" due to the moving conductor across the rail being in two parts ie the rate of change of magnetic flux through the stationary part of the loop and the "top" of the ring is the same as the rate of change of flux through the stationary part of the loop and the "bottom" of the ring.