Since you're just streaming torrents you could just use a torrent client to download the torrent(s) in sequential mode.
e.g. Load the torrent in qBittorrent and make sure to enable "Download in sequential order" and "Download first and last pieces first". When the torrent is about 5-10% downloaded you can start playing the incomplete file while the torrent continues to download in the background. Load the file in your preferred media player (VLC or whatever) and play.