The mesh I came up with is this one. It could be more rounded but just move the supporting edges away from the corner of the hole to do that. I did not round the object as a whole in the corners but this can be done quite easily.
I started with a plane and subdivided it. I then added a loop around where the hole would be and then deleted the faces. I extruded and added some more supporting edge loops. Finally I added an edge loop inside the hole and scaled it to give the bevelled appearance.
Once completed I added a subdiv x 2
Here I added a simple bend deformer to show the flexibility of the mesh for animation etc.
Posting my suggestion as answer in order to show pictures: This is another way to do it.
I inset the face, then moved the result to the adequate position. Then extrude along Y.
Then bevel modifier, add a couple of loopcuts inside the square, apply bevel.
For this one, I did loop cuts vertically and horizontally. Then extruded the square along Y, added a loop cut inside for bevel not to go crazy, then applied bevel.