How does LDS theology suffer when Book of Mormon historical claims are unsupported?

What impact is made within LDS theology regarding the spiritual content of the Book of Mormon when the historical claims therein are shown by both secular and Mormon scholars to be factually unsupported?

None. Just as the bible is not an archaeological/scientific text (garden of Eden, Noah flood, etc), the Book of Mormon is not as well. It like the Bible is a religious text. Archaeological/historic evidence is nice to have but not necessary (though plenty of evidence exists IMO-see related questions mentioned in OP comments). The Book of Mormon's purpose from the title page:

And also to the convincing of the Jew and Gentile that Jesus is the Christ, the Eternal God, manifesting himself unto all nations—And now, if there are faults they are the mistakes of men; wherefore, condemn not the things of God, that ye may be found spotless at the judgment-seat of Christ.

I'll restate: Since we do not know exactly where the Book of Mormon took place, it should not surprise anyone that archaeology and anthropology professors do not believe that it has any support in the New World.

In response to Richard Matheny's quotes:

"I would say in evaluating the Book of Mormon that it has no place in the New World whatsoever."

He clarified this message in "Basic Methodological Problems with the AntiMormon Approach to the Geography and Archaeology of the Book of Mormon," Journal of Book of Mormon Studies 2/1 (1993), pp.190-191 ... he answered as if he were a non-Mormon archaeologist

I received a copy of Heart and Mind and a copy of a letter sent to you by Luke P. Wilson, Executive Director of Gospel Truths Ministries. From these items I feel some obligation to give you a little more information about what took place at the Sunstone symposium in 1984. . . .

I had no idea that I was being used by Gospel Truths Ministries to discredit the LDS Church in their publication. . . . In 1984 I was asked by Sunstone to give a talk, which I refused. They persisted by calling and asked if I would be willing to sit on a panel and comment on papers that would be given on archaeology at the upcoming symposium. To this request I consented. However, when I arrived for the symposium, much to my surprise I was listed as a speaker. I objected and said that I had not prepared a paper. The Sunstone people then handed me a card with a question on it and asked if I would comment on the question. The question dealt with how does a non-Mormon archaeologist evaluate the Book of Mormon in terms of its cultural content and claims. My answer to the question was an ad hoc response where I tried to put myself in a non-Mormon’s professional shoes and talked about the nature of the problems that the Book of Mormon poses for the archaeologist. . . .

Gospel Truths Ministries is using my ad hoc response without my permission, without my knowledge, and in a pernicious way against the church, and against me. The letter sent to you said that a complete transcript of my response was forwarded to you. I don’t know what GT Ministries means by a “complete” transcript. I forbade any publication of my response by Sunstone or any one else, and did not authorize any tape recordings at the time.

_{emphasis mine}

_{also of note is these quotes are from 1984 (~40 years ago) and more evidence has emerged and as it has in the past, emerging evidence won't convince anyone}

Without the Book of Mormon, nothing at all remains. If the Book of Mormon is true, Joseph is a prophet and everything else that entails. If it's not true, then so is the claim that Joseph is a prophet, and everything comes crashing down. It might still be a good spiritual book, but of human origin (and of a human who then wrongly claimed to be a prophet).

Joseph Smith himself stated:

“Take away the Book of Mormon and the revelations, and where is our religion? We have none” (Minutes and Discourse, 21 April 1834, Church History Library, The Church of Jesus Christ of Latter-day Saints, Salt Lake City).

That being said, there are several things that keep "unsupported historical claims" from actually being back breaking.

First, and most obvious, absence of evidence is not evidence of absence. New discoveries are made all the time, for better or worse. For example:

FAIR on the topic of metal plates:

When it first appeared, the Book of Mormon was attacked for the alleged absurdity of having been written on golden plates and its claim of the existence of an early sixth century B.C. version of the Hebrew Bible written on brass plates. Today, however, there are numerous examples of ancient writing on metal plates. Ironically, some now claim instead that knowledge of such plates was readily available in Joseph Smith's day. Hugh Nibley's 1952 observation seems quite prescient: "it will not be long before men forget that in Joseph Smith's day the prophet was mocked and derided for his description of the plates more than anything else."

Second, it's actually unclear where the Book of Mormon exactly takes place. In other words, there is no really specific claim as to the location. That it's Mesoamerica is a theory that is currently the most popular, but it's not the first and only one. Even if a "historical claim" was disproven without a doubt(which I doubt is possible due to the first point), it only really means that a theory as to the location of the events in the Book of Mormon is disproven.

Book of Mormon geography

Since the publication of the Book of Mormon in 1830, members and leaders of The Church of Jesus Christ of Latter-day Saints have expressed numerous opinions about the specific locations of the events discussed in the book. Some believe that the history depicted in the Book of Mormon—with the exception of the events in the Near East—occurred in North America, while others believe that it occurred in Central America or South America. Although Church members continue to discuss such theories today, the Church’s only position is that the events the Book of Mormon describes took place in the ancient Americas.

In conclusion, I would say actually disproving the Book of Mormon on historical grounds seems, to me, not very possible.