< What's strange is - why would the rhythmic, fluctuating idle only occur after the car has warmed up? I think this is the key to the problem. The car must be completely warmed up until it starts happening.>
That indicates fuel management after closed loop .
Another possible is a sticking egr valve, The egr is not activated til warm eng. That would prob. throw the check eng. lite , though..
As a stab in the dark [ seeing the water pump change was when the
fault was noticed] ,you may want to pop the front cover [front of valve cover-plastic] and look at the vac lines to the switch over valves
that run in front of the motor. [ egr and air pump sov's are here]
More than likely your guy has already done so, but worth verifiing
before getting into fuel management, as a vac leak here could cause this complaint.