I think the only way to do this perfectly is to do it as a "Quilt as you go." You could cut it up yet and do this, but you would lose some seam allowance on the back. Maybe you have enough for that.
Cut or open seams top to bottom on each side of the center square. Then cut or open seams to get the center square free from anything else. Now layer the top, back and batt of the center square. Quilt it as desired. Add the top and bottom pieces to the square and quilt that. Then add the side pieces one at a time and quilt them. May be doable but not easy. Could work if your back is large enough to accommodate the extra seams. The seams of the top would be taken apart, so would go back together again. You would probably need to add the top first and work on the back later with each piece.
Does anyone see how this would work?