I think the problem that the OP describes may be that the upright squared wings of the points stick out of the glazing fillet because of the depth of...
I assume the screen frames will not be supported on the sides by the framing of the building because otherwise this wouldn't be a problem. 2x screen...
Yes, if the box slides smoothly and the slides don't have fine adjustment features then just shim the false front. You may regret using brads. Screws...
If I were you, I would borrow or rent a timber framing circular saw and make the cut in a single pass. Then clean it up with sandpaper or handplane....