Are rhumb lines ever used by an FMS as leg paths for RNP/RNAV? If so, what arinc 424 leg type would use them?
No. The FMS does not use rhumb lines.
All RNP/RNAV flight path legs are geodesics. The only exceptions are RF legs (a constant radius circular path about a turn center) and hold legs (a closed racetrack pattern).
To quote RTCA DO-236C, Minimum Aviation System Performance Standards: Required Navigation Performance for Area Navigation:
RNP Routes and user-preferred trajectories are assumed to use a series of fixes. The desired path is defined by a series of geodesic tracks joining successive fixes.
It also includes the following definition:
GEODESIC LINE [Bowditch]
A line of shortest distance between any two points on a mathematically defined surface. A geodesic line is a line of double curvature and usually lies between the two normal section lines which the two points determine. If the two terminal points are nearly in the same latitude, the geodesic line may cross one of the normal section lines. It should be noted that, except along the equator and along the meridians, the geodesic line is not a plane curve and cannot be sighted over directly.