I really love your wall hanging. Did you have a named pattern or your own design? Provided your batting is not too thick, and you're using a quilting/redwork design, then yes - you can embroider through the whole sandwich, and you will not need any stabiliser behind. First you will have to secure the sandwich by either tacking, spray basting or pinning. If using filled designs you would have to do that on just the quilt top with stabiliser underneath, before sandwiching. Make sure your quilt is well supported so there is no drag on your hoop. Please post another photo when done.

I just did a lot of embroidery designs on a queen sized quilt. If you haven't layered your wall hanging yet I'd suggest starching the backing good first (really helps to prevent wrinkles & puckers). On the top mark the centers for your designs. Then sandwich your wall hanging. Do all the SITD first. For the embroidery - Thread your machine for regular sewing, meaning do NOT thread the bobbin for embroidery. Use the same thread in the bobbin as you are on top. This way the design shows the same on the back as it does the front. During the embroidery make sure you take all the weight off the wall hanging, so it doesn't cause any drag on the embrodery arm or hoop.

I use my embroidery machine all the time to quilt my quilts. I do sandwich them first and do not SITD

Don't know about the embroidery part, but that is just beautiful. Love the colors and the design.

That quilt is just so pretty I think I would leave it as it is. A similar quilt hung in the Asheville Quilt Show recently and it looked so pretty just lest as it was.