I have no reason to think it was fake. Why bother? The sync would not be so hard to accomplish: it could have been cued up by use of metal foil on the tape that would stop the machine at exactly the right place.
I also seem to recall noticing in concert that there were very slight variations in when the chorus came in, and thinking that therefore it must be really coming from the tape.
I never heard or thought that it was Carly singing on the tape. It might have been, but I hadn’t heard that.
BTW, did you notice that on the commercial they actively de-emphasize (in the mix, I think) the word “the” (after “shower” and “show them”) that Taylor for some reason sang as “thee.” I always found that pronunciation inexplicable and annoying.