I think what you want to do isn´t really all that doable with native construct, since construct is simply not a 3D engine. I mean it is possible with css/javascript, but at that it doesn´t run all that effective.
codepen.io/edankwan/pen/emqgpr
The closest solution might actually be what you did
A layer with a map image, with a Bulge effect, not so high on the scale as quality gets degraded no matter the image resolution; A layer on top with a white circle. "Source in" on the the bottom layer and "Destination in" on the top layer. This "works" to some extent but it never quite feels like a globe when moving it around with the mouse, it just feels like moving a map around inside a circle
You actually got me curious about that idea, I might give it a shot. I think there might be some tricks I can use (in terms of design/art) to help faciliate a better perception of a globe and not a circle. I´ll post here if I got something. :)