You are making too much of a distinction between “Christian” and “non-Christian.” In this particular usage, the term Christian reflects the presence of a valid baptism. So for instance, imagine there existed a Protestant denomination that decided that abortion was a sacrament and aborted babies on the altar every Sunday instead of celebrating the Eucharist. Absurd, yes, but the members of such a denomination would be “Christian” insofar as the term is being used here and in a way that the members of the Salvation Army are generally not. Is that really a meaningful observation in relation to the subject of giving money to charity? Would you give to this imaginary denomination because they are “Christian” rather than to the Salvation Army? Or, using a more common example, would you give to a Protestant charity with the putative goal of offering material assistance to, say, Latin Americans, when in reality they are primarily seeking to win easy proselytes from traditionally Catholic countries? These charitable groups are undeniably “Christian,” but they are not to be supported.
To me, this discussion of whether Mormons, the Salvation army or other denominations are Christian is incredibly petty. As if baptism was the only thing in the world that mattered. Isn’t it enough that they are striving toward a common good they share with us as best as they understand it? I would give without hesitation to a non-Christian charity that only sought to feed the hungry. Although I understand favoring Catholic charities.