An aveil (mourner) does not practice public aveilut (mourning) on Shabbat (Ketubot 4a), though he is still in aveilut and there is some impact on what he can and can't do.
Would the presence of an aveil in shul at mincha on Shabbat stop the community from saying the prayer Tzidkatcha? He is still an aveil to some degree and not saying tzidkatcha isn't a public expression of his mourning.
Additionally, would his being there even if he is not considered an aveil to that degree still obviate saying tzidkatcha? The rule seems to be "if the day was not Shabbat, would we say tachanun? If not, don't say tzidkatcha on Shabbat." But if the day was not Shabbat, then he WOULD be a full aveil and then we would be davening in a minyan with an aveil and would not say tachanun.