I don't fly too many coupled approaches, so forgive me if this is a
simplistic question.

We were flying a coupled VFR/practice ILS in a Debonair with a
Century-2000 A/P. The heading bug was set about 10 degrees to the left
of where it should have been, resulting us tracking to the left of the
localizer centerline. The CDI was about 1/2 scale "fly right"
deflection but stable.

Had we continued that way, it seemed that we would indeed eventually get
to the threshold, but following an offset course. This would have
required about a 5 degree heading change at DH to line up with the
runway. I suggested resetting the heading bug, which the other guy did,
when we were about 2 miles out. This resulted in the A/P over
controlling, flying us back through the localizer centerline, and then
starting what looked like it would end up being a series of S-turns
through the localizer. We disengaged the A/P at that point and flew the
last bit by hand.

So, my question is, what would have been the right thing to do?

One possibility would have been to do nothing, and just be content to
keep 1/2 scale deflection all the way to the DH. As long as it was
stable, we would have certainly found the runway.

Another possibility would have been to correct the heading bug, but to
do it in smaller increments, moving it a couple of degrees at a time to
let the A/P catch up.

I suppose a third possibility would have been to just be patient and let
the A/P damp out the oscillations on its own, but I think that would
have been a poor idea.