"Directed" mutation has always been looked at in a selective
environment - one in which a substance is present that the
pre-existing cells are unable to use thanks to a genetic mutation,
such as lac-, trp-, etc. So if I understand the question,
gratuitous inducers would have no effect since a necessary quality
of the inducer is that it must allow growth of the cell if a
mutation occurs that allows its utilization.
This is not to say that you couldn't set up a situation
that would allow growth under such an inducer, but to my mind,
(Barry Hall's opinions not withstanding) it wouldn't be
directed mutation per se as has been previously studied.
It should be noted as part of this discussion that a
Macphee published an article in the Feb 1993 Mutation Research
showing that many of the experiments done by Hall and others
are at least dubious thanks to the effects of glucose repression
and the lack of controls for those effects. If you're thinking of
experimenting with directed mutation, I'd give that paper a
good look before proceeding.