sounds like a good idea to me.
Quick story about stains... I had several quilt tops given to me by my MIL. Her mother or grandmother had pieced them. She didnt want them because of the stains. I took them to a lady to appraise them (from what is now the International quilt study Center in Lincoln) She explained that it was out of sheer necessity that they used the stained fabrics.... they couldnt afford anything else. I originally had thought it was stained after it was made, but she pointed out that the stains were in the seam allowances and were used that way anyway. She dated most of them to the thirties... the depression era. It gave me more of an appreciation for them. But I understand your situation with a backing... I think with a backing I might do the same thing. The piecing with stains I decided to leave alone... as to tell the real story of why the stains are there.
Show us a picture when you are done! Wishing you alot of success with your project!
Peggy