Hi across; thanks for responding.
No, the sticker is real -- actually I eventually got a call back from someone at Kansas lab (I never expected to be called back!) and she assured me that even though the form on which the sticker is placed is indeed a photocopy, it's the sticker that the UK is interested in.
She said that they, the lab, keep the original handwritten form, and send back a copy but as long as the copy does indeed have the physical sticker, this will be considered the pet owner's "original."
I was very, very relieved to hear all this! There is a lot of conflicting and confusing information, and I was panicking, but after hearing from the lab I think sure enough what I've been given is what is required. (Whew...)