Having thought about this I'm gonna propose a more technical explanation.
Suppose that for the purpose of authenticating the video, each frame also contains a cryptographic hash signed using public/private key encryption. The upshot of this is that the video can be proven to be created by Alliance personnel. Only they would have the private key necessary to do the encryption. As such the Alliance could not deny the authenticity of the video.
This eliminates the possibility of sharing the video by simply pointing a camera at it. Such a procedure would not capture the hashes, and the Alliance could easily dismiss it as propaganda.
They could read the disk and transmit it. But in order for verification to work, the video would have to be transmitted with 100% accuracy. The video we did see transmitted was probably lossy and of lower resolution. To transmit the Miranda video would require much more bandwidth than any of the video we actually saw.
The bandwidth available to Serenity would be limited. The transmission system probably is not all that powerful. Transmitting a lot of data would probably also attract the attention of the Alliance. Hence they could not feasibly transfer that much data.
TL;DR: the video file was too large to send over the connection.