A Western Army pattern battle flag captured in Virginia was a rare occurrence, however yours never saw a battle unless it was a reenactment. It appears to be made of BullDog cotton bunting and very sloppy sewing to look homemade (or field made).
Check with a local University to see if they have a textile department, that could age the material and what it is made of. If the museum told you that this was original to the war, do you have that in writing? Did you buy it on e-bay?
With the 150th going on, fakes, flags, uniform, guns, etc., are everywhere.
Happy Trails!
Tom Martin