Prism99

I would machine baste the batiste to a pre-washed(and therefore pre-shrunk) good quality muslin, using water soluble thread in the top (can also use in the bobbin if your machine can handle it, but not necessary as bobbin thread will be hidden under the muslin). Baste to a larger piece of muslin (you can iron the two pieces together first), then trim the muslin even with the edges of the batiste. Assemble into a quilt top, baste as usual, then quilt.

Choose the batting carefully. I would probably consider using a Quilter's Dream cotton in either a hand-quilting or machine-quilting weight, depending on how you are going to quilt it. This will keep the batting thin (in keeping with the period) and give the quilt a nice soft drape, compatible with the lawn background fabric. The quilting will keep all of the layers together (including the muslin).

I just think that backing with fabric rather than one of the modern stabilizers is more in keeping with the original blocks.

Edit: On second thought, since this is hand applique, I would probably recommend hand quilting using Hobbs wool batting. This would also be in keeping with the period, and the wool gives particularly good definition to the applique. Although you would be hand quilting through an extra layer (the muslin), it should still be relatively easy to do since both wool and muslin are relatively easy to hand needle.