It appears you already tried a Crease factor on those edges using <kbd>shift-e</kbd>

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 to 1.0 again.