Hmm... can you actually use putty (or any other SSH connection) from outside your LAN? And would you be able to see the CGminer window over SSH?
Yes you can, although you would have to forward tcp port 22 in your router to your BAMT system. Then you'd likely need a dynamic DNS service (
http://dyn.com/dns/ or
http://www.noip.com/) since on most residential Internet connections your public IP address can change. You can see the cgminer window when logged in via SSH by typing "screen -r".
Although if you prefer, BAMT has a RDP server built in. You should be able to connect with just Remote Desktop Connection. You'll likely still need to forward a port (for RDP it's port 3389) in your router and set up dynamic DNS service.