How about something like this:
```scala
  lazy val writeMimaVersion: ReleaseStep = { st: State =>
    val extracted = Project.extract(st)
    
    val (releaseV, _) = st.get(versions)
    val file = extracted.get(versionFile)
    IO.write(file, """scalazMimaBasis in ThisBuild := "%s"""" format releaseV)
    st
  }
// ...
releaseProcess := Seq[ReleaseStep](
  checkSnapshotDependencies,
  inquireVersions,
  runTest,
  setReleaseVersion,
  commitReleaseVersion,
  tagRelease,
  publishSignedArtifacts,
  setNextVersion,
  writeMimaVersion
  commitNextVersion,
  pushChanges
)
```

Not sure if you also need to set the `scalazMimaBasis` for the released version. In that case you'll have to fiddle with it a bit.