Shuls have absolute rights to determine kibbudim of all types. If a shul decides that a non-member has no rights, then he has no rights. His level of “chiyuv” (sic) is irrelevant. If he has yahrzeit or is in shloshim, he must find a stranger-friendly shul. That is the minhag (there is no halachah here, except that the shul has absolute authority) as practiced in Europe for hundreds of years. A good example is the proclamation on the back wall of the Alte-neu shul in Prague from the days of the Noda B’yehudah. Easily available online.
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