Yes, swapping the tape into a new shell is very easy. Just look carefully at the tape path, to make sure you know which bits the tape runs on the outside of and which it runs on the inside of.
Wait: are they claiming it was damaged when you brought it in? If so, but you’re pretty sure it wasn’t, then they are Weasels, and there ought to be some kind of action you can take against them, if only warning all your friends not to patronize them.
If not—that is, if they damaged it—then they ought to do whatever extra is necesssary without charging you extra. (Did they say they would charge extra, or are you just assuming.)
Either way, it sounds suspicious and unprofessional that they would have to pass it along to “a friend.”