Steve,
If I understand your post correctly, the carpet/underlay was still wet when you started.
In this case, unless you spend an inordinate ammount of time extracting, the dirty water from the underlay will keep re-soiling the carpet both from foot traffic and wicking.
Having extracted the majority of the water and lightly rinsed two options are, as you have concidered, renew the underlay or wait untill it has dried completely before attempting the clean.
Since you have left the premises you should base you recommendation on:
a) The shrinkability of the carpet and its' potential for celulostic browning problems.
b) How soon you schedule allows you to return.
c) How confident are you of removeing the rust when dry.
d) How good a carpet fitter are you?
Sorry not a very definate answer but there are a number of variables that cannot be judged at long range.
John.