This case is directly analogous, it seems to me, to the case of opening and shutting a book with words (i.e. a person's name) printed on the edge of the pages. There the the Mishneh Berurah 340:17 says that the custom is like the Rema in his responsa (and other achronim) who rules since this it is intended to constantly be "opened" and "closed" it is not a transgression of erasing or writing. Nevertheless the Mishneh Berurah adds that it is proper to be stringent if possible (i.e. one has another book...or coat).
Nevertheless, since unlike book pages the two sides of the garment are actually joined together as one, there may be even more room to be stringent.