Thanks, GingerK...sort of what I decided on.
I made tubes the exact width of each side of the white square and pressed the seam open so it was on the bottom. I'm going to cut the underneath out, leaving the 1/4" seam on both sides and then Steam a Seam Lite 2 it on, folding over the ends that are open so there are no raw edges. Then, I'm going to FMQ over the block the way it was done originally. I think it will work.
I also Retayne'd the quilt this morning, so I shouldn't have any more bleeds.
Watson