We are now getting reliable HTTP networking with the 3650 by changing the gateway used for Internet access on the phone. In this case from AT&T's mMode ... to an Internet Access Point.
With an Internet access point you use 0.0.0.0 as the gateway IP address.
For AT&T the access point name is "proxy".
Why this is needed, I don't know. It sounds like the firmware can't deal with the default cWap gateway.