Perhaps you can paste it into an image editor app and then save it as a file?
Pretty sure the media picker can only pick files on Android.
Also if the image was on screen at some point before it was on the clipboard, you could take a screenshot as a rough workaround. Those are also saved as files.
I actually took a quick look at the Android API to see if there was an obvious way to make an app to do this, and as far as I could find, there is not. It might be impossible.
Oh! Good find.
Looks like the way to do this is by implementing ACTION_GET_CONTENT. I can't help you beyond that (I don't have a dev environment set up and am a novice anyway)