I think a bigger binding would work. I use the French on were you cut the strip 2 1/2 times the binding width. then you fold it in half, then work with the material double. I stay-stitch the quilt edge before I sew on the binding to give it some oomph.
I like that idea someone had to glue then SITD from front! as I now have 4 placemats and quilt all piled up ready for hand sewing the bindings :(
Good luck - hope bigger binding doesn't affect border too much.
Sue