Here the 1858 Wright edition accidentally omitted the pronoun him. This error was corrected in the subsequent 1874 RLDS edition. We note here, by the way, the occurrence of the nonreflexive him rather than the reflexive himself that is expected in modern English. As described under Mosiah 25:23, the Book of Mormon generally prefers nonreflexive object pronouns. With respect to the verb phrase “to take upon oneself something”, the text has 44 instances with the nonreflexive pronoun and only 3 with the reflexive (Mosiah 15:9, Mosiah 25:12, and Helaman 11:24).
Summary: Maintain in Mosiah 29:6 the earliest text with its nonreflexive him (“and will not take upon him the kingdom”).