I really have problems with this limitation as well. Another part of the Runtime, where curves don't work is the GeoView.IdentifyLayerAsync Method - curves are never in the result set.
The only workaround, I'm aware of, is to simplify the data before using it in the runtime. If you're using the desktop pattern, you can make use of arcpy.Densify_edit() before creating your runtime content. This transforms your curved polygon features to a polygon with lots of interpolation points..
In server workflow, I think there is a server-flag like "SupportsAdvancedGeometry" or similar, which can be set to false to make the server deliver simplified geometries as well.
Hopefully, anyone else comes around with a better workaround that does not involve changing the data beforehand..