While they did execute at least one member of the church, Helmuth Hübener was arrested for giving pamphlets to the French, not his religion, so I do not think religion was a factor.
Nazi Germany was suspicious of Mormonism, due in part to the strong American origins and ties and due in part to their beliefs regarding Jews: that they are a chosen people and that they would be gathered together in Israel. Nevertheless, Mormons joined the military and the church had a policy of non-interference/supporting the government, whereas Jehovah Witnesses and members of other systematically persecuted religions did not. That was a significant enough distinction that Mormons were not systematically targeted. Mormons did face limitations on the exercise of their religion during the Nazi era. The Nazis banned the Boy Scouts in 1935, which was the primary youth program for boys in the church, and instead required boys to join the Hitler Youth, which most young Mormon boys did. In 1939, all foreign missionaries left Germany. During the war, the vast majority of boys and men in the church were involved in the war effort and not available to serve in the church. The Gestapo prohibited distribution of certain church materials due to the teachings about Jews, millenialism, and the gathering of Jews in Israel (Zionism). The Mormons that were punished in Germany faced such punishment for their political actions, as you pointed out in your statement. Hubener, along with three other young men, listened to British short wave radio and were anti-Nazi. They distributed anti-Nazi pamphlets in Hamburg in 1941. For those actions, they were tried for treason and sentenced to either concentration camps or, in Hubener's case, to death (Hubener was only 17 when he was beheaded for his actions). After Hubener's grandfather (who was not Mormon) insisted the church was behind the actions, the Gestapo seized all church records at the local meeting place, but concluded that the boys acted on their own. The Gestapo insisted on Hubener's excommunication, and he was excommunicated by the church by a local leader who was a Nazi sympathizer. That excommunication was reversed later by the church. Due to the members of the church being willing to serve in the military and the church's official policy of non-resistance, the Mormons were not systematically persecuted for their religious beliefs.