So as per your question, your looking for a video editor kind of solution to achieve the task..
I think this your scenario:
** You have a video in your application
** Open a bitmap resource (image file from some where)
** Overlay this bitmap at the bottom of the movie (video) on all frames in the background
** Save the video on external storage
For this ffmpeg does support overlaying functionality or Android MediaCodec stuff. FFMPEG has various filters one of them is overlay filter. What I understand is you want to overlay an image (i.e. png) on the video, ffmpeg surely is a useful framework to do this job. You can set the output format you can set the co-ordinates of the image which is to be overplayed.
E.g:
ffmpeg -i input.avi -i logo.png -filter_complex 'overlay=10:main_h-overlay_h-10' output.avi
Above command adds overlays logo.png on the input.avi video file in bottom left corner.
More information about the filters is available at following website,
https://www.ffmpeg.org/ffmpeg-filters.html#overlay-1
Here is the GitHub links
FFmpeg Android Java
Android Java wrapper around ffmpeg command line binary or this
If this is a solution to your problem you need the C code equivalent to the above command. You also need to see the performance of the ffmpeg because it a pure software framework.
Hope I have understood your question correctly and this helps.