Given the humanist, rationalist, and determinist (remember the influence of Spengler) strains in Welch's writings like The Blue Book, it seems that a conversion to Catholicism, with its doctrines of original sin, a personal, all powerful, and omniscient God, and free will, would be out of character. However, men's minds can and do change. After twenty plus years of troubles and the rise and decay of his organization, Welch may have become more receptive to the message of traditional Catholicism, especially as he faced his own mortality.
It would be interesting to read McManus's memoirs on the matter, if he ever chooses to write them, or perhaps any correspondence from Robert Welch or his wife discussing conversion to traditional Catholicism.