I started trying to find an explanation of your paradox in terms of quantum mechanical amplitudes, but I realised, as you also pointed out, that the problem also occurs classically.
Lets say the source is at rest, and the observer is moving.
The velocity is in the x-direction, and the photons are going in the z direction of the rest frame. So I started writing down the E an B fields for right-circular polarised waves in 2 opposite directions. So you should be able to determine exactly if there is a problem or not. Apart from angular momentum, linear momentum is also tricky. The Poynting vectors in the moving frame are bigger than in the rest frame.