Actually, Spitoon, unless I have misread your post, 'solid CATIII' does not make the problem much worse, since the radalt will be used by the autopilot for the approach.
It is definitely a major emergency for the a/c crew unless VMC/day and is extremely demanding in terms of checking routing and flying at the correct altitude in IMC.
There are, in addition, other little tricks for pilots like the ground return on the radar which can give a coarse indication, not forgetting GPS altitude where available and for both sides, mil height finders..