Fill about a 2.5" gap between drywall frame and and brick
What about just drywalling the faces of the 2x4s and letting it hang off, to fill that 2.5" gap? E.g., use a 6" wide strip. You should probably plywood the bottom sill for safety (yours and the cat's) before you drywall it.

Use corner bead and mud it for a 'temporary-permanent' solution, after you do shove a little more insulation back there.