Is there a keyboard shortcut I can use to set the edge bevel weight?
You can add a shortcut in Edit Mode using the bar at the bottom of the 3D view.
Also notice that the shortcut for the Edges menu is
Ctrl E, so if you wanted you could type
Ctrl E in the 3D view and select
Edge Bevel Weight. You can also add a shortcut from this menu by right clicking and selecting 'Add Shortcut', as above.