The conjunctive but if, in current English, leads to a sentence fragment since there is no independent clause that follows the dependent if-clause. The editors for the 1920 LDS edition substituted unless for but if, which then makes the following clause dependent upon the preceding independent clause. The use of incomplete subordinate clauses is common to the original Book of Mormon text. Many of these fragments have been eliminated by editing (as here), but some examples still remain. See, for instance, Enos 1:1–2 and the reference to it in the discussion under Enos 1:3.
Historically, the conjunctive but if meant ‘unless’, but this meaning is now obsolete (see definition 10b under but in the Oxford English Dictionary). This meaning for but if was especially prevalent from the 14th through the 16th century. The OED lists citations from 1200 to 1596, with the following example from the late 1500s:
In the King James Bible and in Shakespeare’s works, the occurrences of but if appear to have only our modern sense, not the archaic sense of ‘unless’; nor is this obsolete sense of but if found elsewhere in the Book of Mormon text. As noted in the discussion regarding the verb require in Enos 1:18 (where the meaning is ‘request’), the original text of the Book of Mormon seems to have had a number of archaic word uses, usually inconsistently applied, that date back to the 1500s and 1600s. The unique use of but if may be one more example of such usage, despite the fact that it is found only once in the text with the apparent meaning of ‘unless’. For a list of other examples like require and but if, see the discussion regarding ceremony in Mosiah 19:24. For a complete discussion of the archaic vocabulary in the original text, see volume 3.
David Calabro also points out (personal communication) that this particular if-clause could actually be interpreted as a Hebraism. There are some examples in the original text where a conditional clause lacks its expected main clause and must therefore be supplied by the reader; for two examples, see 1 Nephi 19:20–21 and Alma 30:39. Also see the discussion in volume 3 under hebraisms regarding this possibility. Note that the Hebraistic interpretation assumes that the earliest reading in Mosiah 3:19 represents the original text. Similarly, the argument that but if means ‘unless’ (in accord with earlier English) also accepts the earliest reading without emendation.
Summary: Restore the original but if in Mosiah 3:19; the resulting sentence fragment (from the point of view of modern English) can be accepted since the original text had similar examples of incomplete subordinate clauses, including some that are Hebraistic in nature; an alternative way to interpret but if is as an archaic conjunction with the meaning ‘unless’.