Here in the printer’s manuscript, Oliver Cowdery initially omitted the subordinate conjunction that. Later he supralinearly added the that, perhaps when he proofed 𝓟 against 𝓞 (the level of ink flow for the correction is somewhat heavier). The original manuscript is not extant here, and the lacuna between surviving fragments is over five lines long; thus it is hard to tell if the that was in 𝓞, but presumably it was.
It turns out that this is the only occurrence in the Book of Mormon of the expression “X would have that S”, where S is a clause. Elsewhere the text has 76 instances of “X would that S”; there are also four instances of “X would not that S” as well as one in question form, “would ye that S” (in Alma 30:51). The uniqueness of would have that here in Ether 13:2 makes one wonder if this isn’t an error for would that. In other words, perhaps 𝓞 read “wherefore the Lord would that all men should serve him”. When Oliver Cowdery copied the text from 𝓞 into 𝓟, he could have misread the that as have, thus initially writing in 𝓟 “wherefore the Lord would have all men should serve him”. When he proofed 𝓟 against 𝓞, he noticed that he had omitted the that, which he then supplied in 𝓟; but he did not notice that 𝓞 lacked the have (or perhaps he simply neglected to cross out the extra have in 𝓟), so he ended up with “wherefore the Lord would have that all men should serve him”, a unique reading.
The King James Bible has eight instances of “X would that S”, all in the New Testament (as in 1 Corinthians 7:7: “for I would that all men were even as I myself ”). There are also four instances of “X would not that S”, again all in the New Testament. There’s also one example in wh-question form, in Mark 10:36: “what would ye that I should do for you”. But there are no instances of “X would (not) have that S” in the King James Bible. Nonetheless, there is evidence for would have that in English, at least in the 19th century, as in these two examples (here cited with accidentals ignored) from Literature Online :
The first example is a paraphrase of the Golden Rule (Matthew 7:12), which in the King James version reads would that rather than would have that: “therefore all things whatsoever ye would that men should do to you / do ye even so to them”. In addition, Don Brugger (personal communication) provides this example from the Internet, also dating from the 19th century:
Thus it seems that the unique reading in Ether 13:2 is possible, and it will therefore be retained in the critical text.
Summary: Maintain in Ether 13:2 the corrected reading in 𝓟, “wherefore the Lord would have that all men should serve him”; there is evidence for the expression “X would have that S” in 19th century English, although it occurs nowhere else in the Book of Mormon or in the King James Bible.