Just messed around with BT Sync at work. It works! It will require a bit of management, but no more than normal really.
What it does, basically, is installs on your seedbox (wiki on whatbox gave me everything I needed), then you access it via a webui. Add the directory that you want to share, and then generate a "secret" code for it. Now on whatever computer is hooked up to your Synology, install BT Sync there as well, and give it the secret from your seedbox directory. Make a directory on your NAS for it to sync to, and that's pretty much it. You can set that secret code to read only also, so that nothing you do on the computer end will affect the seedbox.
Obviously you are going to want to manage which files are synced, since if you're like me and have a 1.35TB seedbox you don't want to waste 1.35TB on your local device as well, so just have every new download go to the synced directory, and then after it is on your NAS, move it to the correct NAS directory (movies, TV, music, etc.) and then also move it on your seedbox (or delete it). It will determine that it doesn't need to sync that file anymore, and you're all done.
Also, there IS a Synology app that should allow it to function solely on the NAS itself, without a computer running, but I haven't messed with that yet, and since I always have at least one computer running (usually my HTPC NUC which takes almost no power), there's really no point in me going any further. If the power for the NUC went off, the power for the NAS went off, so neither would work anyway.
And since it pretty much says Bittorrent sync, it must use torrents to sync, which means it should get the maximum throughput available, with error correction, so my days of FTP error correction might be over! I'll report back after I've done more, since I've only really messed with it from work so far, not actually done it at home where it matters.