relative speed against other stars is very much detectable by red and blue shift of light. We would definitely know if sun revolves around something, meanining changes its direction.
At these distances, the sun's direction if it were orbiting a black hole would have only changed by a fraction of a degree in the hundred or so years we've been able to measure its velocity relative to nearby stars - is that data precise enough that that is not within the margin of error?
I think we'd see the paralax being all weird as well.
Yes, and I suspect the parallax anomalies would have been noticed a few decades before the redshift / blueshift anomalies: