How do I preserve the filesize of an MP4 edited in After Effects?
In general, it's better to work with lossless, or at least, less lossy formats than MP4 if you're going to be re-editing the video.  If you can, work from a ProRes or Animation codec .mov file of the original file.  

When you export, adjust the bitrate of the video by entering specific bit rate numbers... Vimeo suggests between 10,000-20,000kbps for HD video.  Adjust it downwards until you get the output file size you want, but bear in mind the quality will reduce as the number goes down.