Here the 1837 edition replaced the modal original should with would, with the result that the meaning of the text was altered. With should, the pronoun he most naturally refers to king Mosiah; but with would, the pronoun he refers to the Lord. The 1908 RLDS and 1920 LDS editions restored the original should. The 1837 change was probably accidental.
Summary: Maintain the modal auxiliary should in Mosiah 28:6; the question is whether king Mosiah should let his sons go preach to the Lamanites.