I don't know if this is going to help anyone at all considering Feathers is being utilised in this thread (I'm not using it in the following solution), but I'm going to put this out there anyway...
We are currently creating a social networking companion app using Starling and we wanted to give the user the ability to add emoji glyphs to post comments.
So after a bit of investigation to find the best solution (which led me to this thread) we have decided to implement a combination of using StageText, BitmapData and Starling Images to create an agreeable experience.
Firstly we are adding a StageText instance to the native stage for emoji support and giving it focus - stage_text.assignFocus()
We then use the stage text drawViewPortToBitmapData method to draw the view port to bitmapdata when the instance loses device focus.
You can then use that bitmapdata to create an Image object to add as a Starling Display Object for use within your app.
(We also add a TouchEvent.TOUCH event to the image so the user can tap it to continue to edit by replacing the image with a new StageText object and assigning focus to it upon interaction)
Once implemented it is seamless across iOS and Android devices in our testing.