For something that is going to come off clean and/or leave the jacket unmodified, tape is really your best bet. I would look into poster tape, which is usually double-sided, but just make sure it is the kind designed to come away from the wall cleanly.
Also, I would leave the symbol detached until you are putting on the costume to wear for *insert reason here.* Then attach the symbol. That way, the jacket is already conformed to the dimensions of her body, thus the symbol will follow suit when attached, and pulling/stretching of symbol/jacket will be reduced.