Has anyone cycled the Tunnel Route?

I am considering cycling from Irun to join up with the Camino Frances at Burgos or Santo Domingo de la Calzada. According to the co-fraternity of St James web site the tunnel route is suitable for cycling. Looking at some of the pictures available for the approach to St Adrians Tunnel. It is obvious that you would not be able to cycle some of the steep sections. However I don't mind pushing/carrying my bike for short distances. If anyone has done this route, do they think this is possible?