It sounds to me like your ISP isn't allowing connections to live very long, even if they're using websocket technology (what our webclient uses, whether in lockmode or not.). They're granting slightly longer connections to SSL traffic, but they're still killing the connection. You don't have an Internet Service Provider, you have a Web Page Service Provider who only wants you to visit websites and stream videos. You have an asshole for your ISP.
A VPN might be your only solution in this situation.