Here the 1858 Wright edition replaced his with this. This secondary reading continued in the RLDS text until the correct his was restored in the 1908 edition. It seems rather odd to refer to “this church” in this passage, although it is not impossible. There is no nearby instance of this that seems to have triggered the error; what we apparently have here is a simple misreading of his as this. About 70 percent of the mix-ups of this and his show this changing to his (for three examples where the change has persisted in the LDS text, see under Alma 9:19, Alma 23:3, and Alma 51:10). For each case of mix-up between this and his, we follow the earliest reading, thus his here in Moroni 7:39.
Most commonly, the text reads “the people of the church” (14 times), but there is one other instance with his (“the people of his church”, in 2 Nephi 25:14) and also one with my (“the people of my church”, in 3 Nephi 18:5). There happen to be no instances with this; even so, the demonstrative is possible since there is one example with that (“the people of that church”, in Mosiah 26:33).
Summary: Maintain the possessive pronoun his in Moroni 7:39 (“among the people of his church”), the original reading; his church works much better here than this church.