The original manuscript is not extant for any part of the text near this passage. Oliver Cowdery initially wrote “with exceeding joy” in the printer’s manuscript, then corrected the text by supralinearly inserting great with no change in the level of ink flow. Elsewhere in the text there are ten occurrences of “exceeding great joy” and three of “exceeding joy”, so either reading is possible, but the use with great is favored. Since either reading is acceptable, Oliver was undoubtedly correcting to the reading in 𝓞. (The 1981 LDS edition changed all these instances of “exceeding great joy” to “exceedingly great joy”; for discussion of that editing, see under 1 Nephi 2:16 or, more generally, under exceeding in volume 3.)
Summary: Follow the corrected reading in 𝓟 for Alma 62:1 (“with exceeding great joy”), which was undoubtedly the reading in 𝓞 (no longer extant here).