With quilting an inch away and that we usually use 1/4" seams, you still have quite a bit to work with.
Some times when we sew on vintage fabrics, especially those that have been used, our stitches themselves can cause issues to raveling/deteriorating fabric. Hope this makes sense without a picture -- I wouldn't necessarily trim down to really good fabric, but maybe I'd have 1/4" of looser woven/raveling fabric above my binding, stitching in 1/2" from the edge. If the fabric can take it, maybe use a zig zag or a fancy ending stitch, either in a one time around or sew your normal seam and then go back around again with the zig zag. When you close the binding, just fold in the bit of extra you have there.