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.
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:
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?