On most windows, once you change the template to WDT_RoundedNoTitle, and set the titlebar to false, you can no longer drag them. You have to move them by re-sizing.
I suspect the chat window is draggable after adding the ClientMovable tag, because that window has 2 inner windows and the space between the inner and outer windows allows the dragging.
You could try creating an inner window in the container window, and moving the slots into it. (i'll test this when i get a few minutes)
Or, you can define your own window with an invisible titlebar.
That is what i did on my UI
Look at the group window for an example.
template is WDT_RueNT and the titlebar is set to true