The original manuscript is not extant here, but spacing between extant fragments suggests that the subordinate conjunction, if it was in 𝓞, was inserted supralinearly. The printer’s manuscript and all the printed editions have the that.
There is some evidence that Oliver Cowdery occasionally omitted the that after “it came to pass”. For a clear example where he omitted the that as he copied the text from 𝓞 into 𝓟, see under 1 Nephi 7:7. But there is also evidence that Oliver sometimes added the that after “it came to pass”, at least momentarily; for an example, see under 2 Nephi 1:1. Oliver could have intentionally added a that here in Alma 57:7 in order to eliminate the awkwardness of “and it came to pass it was our desire to wage a battle”.
As discussed under Alma 55:14, there are a couple of instances in the earliest text where there is no that between “it came to pass” and an immediately following main clause. Theoretically, the original text could have read without the that here in Alma 57:7, despite its awkwardness. There are seven other occurrences in the text of “it came to pass that it …” but none without the that. Here in Alma 57:7, the critical text will follow the earliest extant reading, namely, the reading of 𝓟 with the that.
Summary: Accept in Alma 57:7 the reading of the printer’s manuscript: “and it came to pass that it was our desire to wage a battle”.