You seem to indicate that you can't connect Commander or a web browser at the same time... or connect and ping at the same time. Actualy, when a web browser or Commander is "connected", there is no actual TCP connection to Intel AMT since a connection is established just when a request/response is made.
I am going to take a wild guess. Using Commander or a web browser, log into Intel AMT and change the power settings of the Management Engine (ME) to "always on" and put the idle timeout to 0. This will insure that AMT never goes to sleep. I suspect that what is happening in your case is that AMT goes to sleep and takes about 20 seconds to take-up again when you try to access it, or ping it. If you try the web browser it will fail (but cause AMT to wake up) seconds later you try Commander and it works, etc.
If you can't ping the AMT computer, it's a good indication AMT is asleep.
Let me know if this is it.
Ylian (Intel AMT Blog)