The problem with the WebClient Remote Console is that it doesn't connect through the WebClient Server as one would think, but instead connects directly to the ESXi host the VM is running on (on Port 902). Very clumsy and inconvenient when you want to isolate ESXi management traffic while still allowing remote console access for users.
If you check netstat when you try to connect on the Client, you will probably see some SYN_SENT connections to the IP of the ESXi host on port 902 (if the client can resolve the IP of the hostname).
Since a VMware-proprietary protocol is running on that port (and not within a generic, proxiable SSL-tunnel), you can't access it through Webproxy either (the Console Plugin/Browser will ignore Proxy settings due to that).
vCould Director includes a nice Remote Console proxy to bypass this, I wish VMware would implement that for the WebClient as well