Hey Blendash,
When you apply a boolean modifier you will often have to clean up the topology to match it with the rest.
What you can do is apply it and then merge the vertices from the cutout shape that fall into a square to the nearest of the sphere’s vertices. (Left to middle shape) I find it the most convenient to work only with the cutout and later extrude the cutout inwards. Afterwards you can grid fill the extruded part to match the surrounding geometry. (Right shape)
With that in place you should be able to apply a nice bevel. I hope it helps.
In my example I used a non-destructive Bevel modifier so it’s easier to adjust afterwards.
