Sorry, there is no way to register your app to display in the Share Picker for video files.
We tried to do the same, with no success, and the same goes for photos other than JPEG files, like for example PNG.
To add a quote from the article posted by @ahandersson on "Extending the share picker for Windows Phone 8":
This topic describes how your app can appear in the share picker, a page that lets the user pick an app for sharing the photo they are viewing. Note that this extensibility is only available when the photo is a JPG file.