I've used facings sometime instead of binding. It doesn't have to match the front because you attach your folded strip to the front side, then fold it to the back and hand stitch it down back there.

The bindings on my quilts rarely are from the same fabric that I used in the quilt. I wait until the end and then audition several different fabrics and then when I decide cut my binding from that. I generally cut it 2 1/4 but sometimes 2 1/2 and join the ends with a 45 degree seam. I usually gravitate to a darker or busier fabric so will will show the dirt less as the edge of a quilt usually gets the most wear. Relax, you probably have something in your stash that will work.

If we could see the quilts it would be helpful. I have taken all the scraps from a quilt and made binding from them. I use random length strips cut @ 2.25" wide and sew them together to make the binding.
Hope you figure something out and please show us your finished work.
peace

I'll be doing one like this soon and was wondering if I need to do any kind of stay stitching before I fold it to the back so it will lie flat.

For the facing, after it's attached to the front, I go back and try to trim it down before folding it to the back. The corners especially are bulky, so I trim those down quite a bit. Stay stitching probably would work. I've not tried that, but it certainly worked when doing clothing facings.

I would definitely not use ribbon. Here ribbon is usually made from fibers other than cotton. Unless it is just going to be a wall hanging the ribbon binding might not survive the first time you launder the quilt. The facing idea would work especially if you can't match the colors enough to suit yourself. It would be invisible from the front.

"Or do you have enough of the back fabric to roll up over the top

I have used left overs from other quilts all sewn together. I really like that look.

Oh crud. That's what the extra yardage was for. I just did the same thing you did. I made a second lap sizedquilt to match the bed quilt I'm still deciding how to quilt because I had so much leftover. Used every last stich of the one fabric and was proud of myself, til I read this and then the horror set in.

A friend always told me that black binding matches every quilt. I do not necessarily agree but have used it in a pinch when necessary.