How to calculate the radiation pressure on a moving surface?

Good question. First it depends if surface is like perfectly reflecting mirror or absorbing black surface or other. (Reflecting surface gives more pressure up to double that from absorbing one, as with elastic or inelastic collisions.) Next We would normally expect the motion of real surface to be negligible compared with speed of light and just use usual radiation pressure. A small correction term to this may be what you want. A simple non relativistic Doppler shift will give a slightly reduced frequency and different radiation pressure which should be good approximation. Perhaps to be more precise consider a change of reference frame and move along with the mirror and now the incident wave will be doppler shifted, and we can even use the relativistic result. Then calculate the radiation pressure for this new case, then transform this force back to the original frame using the relativistic gamma factor arising in the relativistic 4-vector for force. Note transverse area for pressure is invariant. This should be good if reflecting surface isn’t accelerating too much as consequence of said radiation pressure which would normally be the case. One could alternately use photons and elastic reflections. This problem is probably somewhere in Jackson “Electrodynamics” or Panofsky Phillips” or other text.