The 1888 LDS large-print edition replaced “they should let no pride … disturb their peace” with “they should not let pride … disturb their peace”. The occurrence of let no here is unique in the Book of Mormon text. On the other hand, not let occurs in three places:
Note, however, that in each of these three cases, there is a preceding form of the auxiliary verb do (which is not the case in Mosiah 27:4). The safest solution in this situation is to follow the earliest textual sources.
Summary: Maintain the original reading in Mosiah 27:4: “that they should let no pride nor haughtiness disturb their peace”.