With respect to my fellow QB members, I would not necessarily use fusible interfacing for this project. Reasons:
Even the featherweight stabilizer makes fabric stiff. About the only one that doesn't is Mistyfuse and you are talking a lot of $$ to fuse misty fuse to that.
The quilt top appears to be out of square and has some fullness to it. Totally understandable as it was a pillow, then taken apart, then washed and the fabrics were thin too. Applying a fusible interface will be tricky to avoid pleats and puckers being ironed into the fabric. You would almost have to cut small pieces to match every piece of patchwork if you were going to take this path. IMHO this would be an exercise in frustration and there is ZERO room for error because once you fuse it is pretty much fused for good.
The best advice I saw here is to find a quilt shop, NOT JOANNES, but a real quilt shop and ask for help from the employees. Bring the tops with you. They will be able to actually evaluate the condition of the fabrics and advise if it is even feasible to make these into lap quilts or simply go with the framing idea.