The original manuscript is not extant here. Initially in the printer’s manuscript, Oliver Cowdery omitted the subordinate conjunction that, but virtually immediately he inserted it supralinearly (there is no change in the level of ink flow). His correction undoubtedly reflects the reading of the original manuscript. As discussed under Alma 21:10, the text is fairly evenly divided between “it came to pass that as …” and “it came to pass as …”, so either reading is possible here in Alma 62:15. Thus there would have been no motivation for Oliver to edit the text either way in this passage. For a similar case where Oliver initially omitted the that in this expression, see under Alma 62:28.
Summary: Accept in Alma 62:15 the corrected reading in 𝓟, the subordinate conjunction that before the as-clause (“it came to pass that as …”).