I think your best bet is to use stereographic projection to map the problem from the sphere to the plane. This projection is very useful in that it maps your rhumb line to a logarithmic spiral of form R=exp(θa) (aka a loxodrome) and maps your great circle to a circle on the plane. So your problem reduces to finding the intersection of a logarithmic spiral and a circle (after which you map back to the sphere).
This is just a sketch to get you started. If you need more details, just ask.
