I really don't know; sorry! I would probably start with Rat's fork of Karl's Perl repeater and test. I know they support at least mode II, and might support III too, since they're based on the Windows code.
See also a thread I just started, which right now is just a reformatting of the above list: All VNC Repeaters
Another thing to consider -- assuming you control at least one side of the equation (the viewer) couldn't you use mode II? That is, you can allow all the various "servers" to connect on 443 (set it as the listening port on the repeater) while you as the viewer can connect on another port.
Also, you might want to keep an eye on the https proxy experiments:
ChunkVNC_https - proof of concept
ChunkVNC 3.2 HTTPS Test