Lightbrite – there is a very common misconception regarding the halachic definition of the terms Kiddush Hashem and Chillul Hashem.
A lot of people think that anything that causes not-religious Jews to think well of Religious Jews or non-Jews to think well of Jews is by definition a Kiddush Hashem, and that anything that causes not-religious Jews to think badly of Religious Jews or non-Jews to think badly of Jews is be definition a Chillul Hashem.
However, that is not correct according to Halacha. If someone is doing the wrong thing, even if looks good to others, it is a Chillul Hashem. Likewise, if someone does the right thing, even if makes others look down on him, it is actually a Kiddush Hashem and not a Chilul Hashem.
Of course, if someone can do the right thing in a pleasant, polite way that doesn’t look bad, then that is the better choice and a bigger Kiddush Hashem. But if it is not possible, then the biggest Kiddush Hashem is to do the right thing anyhow.
There may be some cases where one is supposed to “compromise” for the sake of a Kiddush Hashem, but that is generally only in a case where you are not actually doing something forbidden. Maybe you are giving up on a chumra, for example, because it will offend someone. Or in some cases, you may be allowed to follow a leniency in order not to offend someone. Those are the types of things that can come up when dealing with not-Religious family members or others, and one should ask a sheilah in those cases.