Unfortunately there isn't really a good way to do this via the Dropbox API, but I'll pass this along as a feature request. I can't promise if or when that might be implemented though.
Currently, you could retrieve the video data via /2/files/download, but you'd need to pass that through your server anyway to avoid exposing your access token client-side.
Alternatively, you could technically get a shared link for the file using /2/sharing/create_shared_link_with_settings, and modify it for direct access so you can pass that to a video player. That would be subject to bandwidth limits though.
