HALACHA-----mechalel shabbos for akum?

You’re allowed to save a goy on shabbos, you just can’t be mechalel shabbos to do so. But you are more correct about the "chilul hashem" than you even realize -- there is a heter brought in the poskim that if the akum realizes you didn’t save him then they will have the same attitude towards us when we need to be saved, and worse....

So the heter says to save the akum on shabbos anyway, in order to be able to save the life of the Yehudi which may be endangered later on.

The halachah of not saving the akum is therefore only narrowly applied, since in most cases, you would be correct.

How can we possibly know whether or not refraining from being mechalel shabbos to save a goy's life will result in a backlash of anti-Semitism (Ch"v) that will cause loss of Jewish lives?

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We can’t know. That’s why we don’t do it - because it may lead to a backlash.

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So it should be clear that even though l'maaseh, as I said many times, we do save the life of Akum on Shabbos even if it means chilul shabbos, but that is not because saving the life of Akum supersedes Shabbos - it doesn’t. Rather, because we are afraid that the life of Yisroel will be in danger due to a lethal backlash of anti-Semitism, therefore, in order to save the lives of Yisroel that would be in danger due to that backlash, we therefore are mechalel shabbos to save Akum.