One way to put it is: You are basically asking which is worse, Chillul Shabbos etc or Avodah Zarah?
Answer: No doubt, AZ. It’s basis is found in the Torah and Chazal etc.
The main issue with Chillul Shabbos is, it is considered “AS” a mumar etc. being that Shabbos “signifies” Amunah. (which is debatable and depending the situation etc), hence, official declaring one as a Mumar and a Meshumad is the lowest you can get.