The Labeling API is not yet supported. You can see where it falls on the Parity list here: https://developers.arcgis.com/flutter/parity/
It's further down near the bottom. I'll make note of your interest in Labeling in our internal tracker. If you'd like to see it prioritized higher, you could discuss your needs with your account manager or our product management team (or here). Describe your use case, how essential the Labeling API is, and when you plan to ship. That will help us balance priorities.
