It appears you already tried a Crease factor on those edges using shift-e
https://docs.blender.org/manual/en/dev/modeling/meshes/editing/edges.html
This edge property, a value between (0.0 to 1.0), is used by the Subdivision Surface Modifier to control the sharpness of the edges in the subdivided mesh. This operator enters an interactive mode (a bit like transform tools), where by moving the mouse (or typing a value with the keyboard) you can set the (average) crease value of selected edges. A negative value will subtract from the actual crease value, if present. To clear the crease edge property, enter a value of -1.
I wonder if the bevel operator isn't propagating the crease property. Try applying the bevel modifier and then set the crease again.