With apathy

Simply, most non-LDS Christians would not consider the argument to be persuasive enough for consideration. That doesn't meant to say that it isn't useful at all, but just that it doesn't seemed to be targeted at non-LDS Christians.

The Logic Itself

First, let us examine the argument in isolation. Would the argument itself be a sound argument on its own merit? I do not think so. For this argument to hold, it needs to assume several things which do not seem to be true:

1. It would not be possible for one author to write in the style of multiple authors.
2. Multiple source authors presumes they must have been authentic authors.

Regarding the first point, while stylistic study may help later readers understand and distinguish between multiple voices and styles of writings between known or at least honest authors, those same methods would not be as reliable when considering dishonest authors who one may reason have the incentive and ability to take on various writing styles. Even some reputable and genuine authors may inadvertently have divergent writing styles when writing different types of documents. The first anecdote that comes to mind is how Lewis Carrol's mathematical texts were called "dry". Imagine how much more pronounced this would be if one had intended deception or somehow suffered delusion.

On the second point, even if there were multiple authors, there is no reason to believe that these authors must have been authentic. We have no alternative records to know what style the alleged original authors would have used as we have no separately verified documents with which to compare. There is also no need that I can see to assume that these works were compiled over any small number of days. The claim that the work had not been done elsewhere seems to rest entirely on Joseph Smith's claim, and perhaps one or a few other individuals at most. It doesn't seem implausible to me that they could have been various works written even by multiple contemporary authors that were presented to Joseph Smith by the time he claimed to have received them divinely.

So, in general, I think that many non-LDS, whether Christian or not, would probably respond to the claim with mostly apathy. At best, it seems to perhaps give somebody pause to consider Mormonism a con by a single individual, but not even to rule that out completely.

The Religious Perspective

The Question of Cannon

While it can often be hard to generalize for all non-LDS groups that self-describe as Christian, I believe that for this question, it is a little easier. Historically, most popular branches of Christianity, whether Catholic, Protestant, Evangelical, or otherwise, they seem to be in common in generally agreeing that there are some very unique requirements for a text to be considered to be Scripture. While there have been and still remain many disagreements between some of the most distant branches, the whole body of works within those disagreements are far more similar to each other than the Book of Mormon, and the divide between them considers neither style nor number of authors when making that determination.

The Old Testament and the New Testament, and then also the Apocrypha, have slightly different rules for inclusion, but it seems that, regardless of how a denomination sits on these matters, they tend to agree that two large factors are:

1. Widespread adaption of contemporary believers
2. Consistency of theme

These are typically the playing field when cannon has been debated. Those who argue for adding a work to the cannon would argue that the work was written and copies were indeed passed around extensively to the local congregation(s) of believers at the time and that the theme is consistent with the rest of the work. Neither style nor authorship contribute to these. In fact, we know that there are works written by Paul which are not cannon. Furthermore, there are works, such as The Epistle to the Hebrews and also Ecclesiastes with uncertain authorship. In those works, there is much discussion about style, but never as a grounds for changing status in the cannon.

The Book of Mormon still fails to be a document which was widespread among the pre-existing body of believers at any time (either upon its writing or upon its alleged re-discovery), and many (or perhaps most?) non-LDS would see many thematic differences between that work and any cannon of Scripture that they currently accept, and the argument presented here does not address that.

If one is being fair, they may point out that the Old Testament, or perhaps the Torah or Tanakh, had different rules for what the congregation of believers was at that time, and one might even claim that the New Testament makes thematic claims or changes that the Tanakh did not make. Also, there are some fringe groups in modern day who consider newly discovered documents to be available for inclusion in cannon. In that since, you might argue that multiple authors hints at these being the same as discovered works. I think that such arguments might be a stretch, but they might be valid. At that point one has to consider the reasons why non-LDS Christians are Christian in the first place. These reasons (from a secular perspective) are as varied as the individual, but many times they are often based upon things such as archeological evidence, prophetic evidence, philosophical or moral arguments, scientific claims, or even occasionally social or cultural reasons, and these would not likely be altered by a technicality, nor would an argument about style or authorship be likely to have such an effect.

Differences With Scripture.

Additionally, non-LDS Christians typically reject the Book of Mormon on the grounds that its alleged discovery is suspect, the lack of verifiable claims, and also that the doctrine contradicts other Scripture.

For all other texts considered Scripture by non-LDS Christians,they were all delivered directly, through inspiration of God through holy men, writing in public, making presently falsifiable claims, often even involving miracles. There seems to be no Biblical precedent for God hiding His Word or letting people obscure it. As such, it seems odd to such Christians to think that such a God would allow a new message to come through one man and a secret process. The argument does not address this concern.

Non-LDS Christians often champion the historical reliability of the Bible. The places, events, and people are often discovered through archeology, and the claims made tend to hold up even in opposition. We can see on the map places from the Bible. We can see secondary sources citing events. We can talk often about how and when prophecies were fulfilled. Contrary to this,the events, people, and places described in the Book of Mormon seem to either lack evidence or simply defy what we learn about the area. While that may not be the best proof of inauthenticity, it is a common contention and this argument seems only tangential.

The biggest problem, though, is that non-LDS Christians see the Book of Mormon and the other writings of Joseph Smith as contradicting Bible doctrine. For instance, they believe the Bible teaches that God is the creator of all things and eternally existent, and that Jesus is the same God, eternally existent and always God. They believe that the Bible teaches salvation through faith in Jesus. They believe that the Bible teaches that the Gospel is complete and that the next event will be Jesus' return. Therefore, when they learn about the claims of Mormonism, they see it as a false doctrine. Nothing in the argument addresses that concern.

Overall, the argument seems to be less effective in convincing somebody of the authenticity of the Book of Mormon, especially somebody who is a non-LDS Christian, but rather to provide an emotional appeal to people who are already convinced of its authenticity, and probably best as a part of a network of similar appeals.