Your coat is wonderful!
A lightweight fusible is a possibility. I can give you a brand that is lightweight and often used in silk tie quilts, but it will have to be later today when I can go find the bolt. The biggest issue with using a fusible is the possibility of melting the fabric, as fusibles require heat. What you would want to do is purchase just a yard of the fusible, use a piece of muslin or other fabric as a pressing cloth, and test the fusible on *each* of the fabrics before committing to it. This means you need to cut a test piece out of every one of the fabrics. It is the only way to know if the fusible will be safe for every fabric, and it will prevent you from ruining a large piece of fabric. What a fusible interfacing does is add stability to fabrics that tend to be slippery and/or stretch. It also adds substance to a fabric that is very thin and lightweight.
Using a heavy starch is an alternative way to stabilize the fabric so you can piece it. Because some of the fabrics are very thin, I would also line the entire quilt top with lightweight muslin before sandwiching. You would want an inexpensive, very lightweight muslin and you would want to wash and dry it twice before using to make sure it has shrunk completely. I would not line each block with muslin. Instead, I would wait until the entire top is finished, then spray baste the muslin to the entire quilt top. Then sandwich as usual and quilt. The quilting will permanently join the top to the muslin so that the fabric is stabilized even after you wash out the starch.
As long as all the fabrics used in the quilt have survived washing on their own, it's fine to mix the fabrics in a quilt. Because of the differences in fabric weights (some heavy, some light), I would always hand wash and lay the quilt flat to dry.
If you want a utility quilt rather than a show quilt, you might want to consider making two quilts -- one with all lightweight fabrics, and one with all heavier weight fabrics. This is because these fabrics will wear differently when subjected to actual use (such as sitting on the quilt, wrapping it around the body, etc.).