I was searching for the same issue and saw your post, I am not sure if you are still looking for a way to airdrop your MP4 file to IPhone/iPad into anything other then the Photos app.
I did it by rename the .mp4 file to .pdf file prior Airdrop it to your other device, by tricking the system thinking the file is a pdf file you then have an option to open in the 'Documents 5' app. An error message will prompt to tell you the file is corrupted since it is not a pdf file, just go back to the Documents inbox, select your file and rename it back to .mp4. You then can 'open in' any other apps you wish!
Is not ideal method but at least it get the work done.
12-07-2015 07:10 AM