The original manuscript is not extant here. There is no repeated the in the earliest textual sources, but the 1902 LDS missionary edition accidentally inserted the the. This secondary the was not transmitted into any subsequent LDS edition since the 1902 edition never served as a copytext.
Generally speaking, “cause of X” is preceded by the (47 times) or a (3 times). The only time it is not immediately preceded by a determiner is here in Alma 62:1. In conjunctive phrases, when the final noun conjunct is “cause of X”, we get the before cause (except, of course, here in Alma 62:1). There are two examples:
But in each of these examples, the preceding noun conjuncts take the determiner our; thus omitting the determiner before cause in these two passages would create an anomalous conjunct (as in “we will maintain our religion and cause of our God”, as if the text read “we will maintain our cause of our God”).
We should also note that there is another conjunctive example where the first conjunct is the freedom (as here in Alma 62:1), and in that case the the is not repeated for the following noun conjunct:
So it seems possible that the the does not have to be repeated in such conjunctive phrases. The critical text will maintain the earliest extant reading in Alma 62:1 (“that he was not also a traitor to the freedom and cause of his country”), that is, without a repeated the before cause.
Summary: Although the lack of a repeated the before cause in Alma 62:1 (“the freedom and cause of his country”) may seem a little odd when compared with the normal Book of Mormon style, there is another example where the noun conjunct following the freedom does not repeat the definite article (in Alma 60:36, which reads “the freedom and welfare of my country”).