Your 2011 Augusta is almost identical to my 2009 BH3670. Are you certain the blinds are attached to the top valence and not screwed to the wall just above the window? My day/night shades were screwed to the wall. When I removed them to install vertical blinds back there, I first had to remove the screws that held the side valences to the wall and the top valence, then the screws that held the top valence to the underside of the cabinets above. Then I could remove the screws for the blinds.
But, for your dilemma, there is 1" x 1" framing under the cabinet bottom along the rear wall. You might be able to drive some screws through the top valence and into the framing. Run the screws at a slight angle no more than an inch from the wall. Use an 1 1/4" wood screw.
I know there is small framing in there because I used it to mount my rebuilt top valences for the rear windows and the ones in the main slide to accommodate the vertical blinds.
An alternative to secure it would be to use some longer machine screws, and drill holes straight up and through the cabinet bottoms and through the floor of the cabinet. With washers and nuts, you could then secure it. I used this method to mount a towel rod under the cabinet above our toilet. The screws don't have to be large diameter, something like a #8 or even a #6 would do, just use a large diameter washer for better support.
It's entirely possible that when they mounted the top that they just ran the screws into the thin plywood underside skin of the cabinetry and missed the frame. A bit too much torque and it's destined to work loose.
Sorry for the long winded detail, but I don't know what your "skills" are.