I think your backdrop fills the whole TopLevelControl and blocks mouseinput to it, remove the mouseEnabled="true" from it and it should become movable as long as you don't click on the area the label takes. Clicking on the label will do the other stuff.
You also don't need movable="true" on the backdrop. The TopLevelControl itself is movable so everything in it moves with it.
Edit: If the backdrop is supposed to fill the whole TopLevelControl you could also use <AnchorFill /> instead of dimensions.