You are misquoting the Ran. The Ran does not simply say that any religion which bows is Avodah Zarah, even if they are bowing to Hashem. Besides for the fact that that wouldn’t make any sense – because we do that too on RH and YK – if you read his words you see it’s not what he means. He writes:
?????? ???? ??????? ?? ????? ??? ?????? ?? ????????? ??”? ???? ????? ?????? ?????? ????? ????? ???????? ?????? ??????? ?? ????? ??? ?”? ?? ??? ??? ???, ??? ????? ??? ?? ??????? ?????, ???? ????? ?????, ??? ?????? ????? ?? ????? ??? ??????
He is basing his ruling on his belief that the Moslems bow to Muhammad, and not just a bowing of respect but a bowing of ?????. His whole proof that they are bowing a bowing of ????? is because you don’t bow to a dead person simply out of respect. It is clear, therefore, that his issue with the bowing is the fact that they bow to Muhammad, who is dead, and if they would not bow to Muhammad, his whole vort would not start.
The Moslems do not bow to Muhammad.