According to the LDS Church, is the Book of Mormon a translation from a foreign language?

Yes.

If so, what foreign language manuscripts did Joseph Smith use to produce the Book of Mormon?

Engraved golden plates buried circa 500 AD by Moroni, who returned as an angel to instruct Joseph Smith to retrieve and translate them through inspiration.

He called me by name, and said unto me that he was a messenger sent from the presence of God to me, and that his name was Moroni; that God had a work for me to do...

He said there was a book deposited, written upon gold plates, giving an account of the former inhabitants of this continent, and the source from whence they sprang. He also said that the fulness of the everlasting Gospel was contained in it, as delivered by the Savior to the ancient inhabitants;

Also, that there were two stones...and the possession and use of these stones were what constituted "seers" in ancient or former times; and that God had prepared them for the purpose of translating the book.

Joseph Smith History 1:33-34

Mormon, the eponymous compiler of the record, says they wrote using a modified form of Egyptian, which was shorter to write than their native Hebrew

And now, behold, we have written this record...in the characters which are called among us the reformed Egyptian, being handed down and altered by us, according to our manner of speech.

And if our plates had been sufficiently large we should have written in Hebrew

Mormon 9:32

Oliver Cowdrey assisted Joseph as a scribe, and described the translation thusly:

These were days never to be forgotten—to sit under the sound of a voice dictated by the inspiration of heaven, awakened the utmost gratitude of this bosom! Day after day I continued, uninterrupted, to write from his mouth, as he translated with the Urim and Thummim, or, as the Nephites would have said, ‘Interpreters,’ the history or record called ‘The Book of Mormon.’

Messenger and Advocate, vol. 1

As your question alludes to, the role of Joseph Smith as a translator is different from the usual treatment of the Bible, wherein only the original authors are venerated as inspired prophets of God.

With the Book of Mormon, inspiration is attributed both to the original authorship (by Nephi, Mormon, Moroni, etc.) as well as to the miracle of translation by Joseph Smith.

Lost in translation

This is a point that I pondered, in drafting the question, of your reference above. I could not think of a word or phrase, that would allow for the definition I accepted in this question.

I feel obligated to clarify this here. I'm sure you are able to appreciate the difficulty I encountered, in attempting to phrase my own question with the word 'transmission'.

Joseph Smith often used the words "translated" and "translation," not in the narrow sense alone of rendering a text from one language into another, but in the wider senses of "transmission," having reference to copying, editing, adding to, taking from, rephrasing, and interpreting. This is substantially beyond the usual meaning of "translation." ...

Amend: The above point, I am making with the quote, from the user whose answer I accepted, in the same post; is not inextricably linked to the context of Bible translation, Book of Mormon translation, Pearl of Great Price, or any other specific document. It is a general statement made by the individual who claimed:

"Joseph Smith often used the words "translated" and "translation," not in the narrow sense alone of rendering a text from one language into another, but in the wider senses of "transmission,"

I cannot logically see how the context of translating 'reformed Egyptian hieroglyphs' by the use of a 'seer' stone can be excluded from Joseph Smiths definition of 'transmission'. If I am wrong here somebody, please, correct me. I was accused of taking this out of context.

This is one of the last testaments of Emma Smith on the 'translation'. I must also point out here, that Emma Smith says "...it was impossible for him to see how I was writing them down at the time."; to clarify, that Joseph Smith was viewing through the stone, and reciting to her as his amanuensis.

"When my husband was translating the Book of Mormon, I wrote a part of it, as he dictated each sentence, word for word, and when he came to proper names he could not pronounce, or long words, he spelled them out, and while I was writing them if I made a mistake in spelling, he would stop me and correct my spelling, although it was impossible for him to see how I was writing them down at the time." (Emma's statement, in an interview with Edmund Briggs in 1855)

There is no way of relating the term translation in any traditional sense of the word, with the method described by Joseph Smith himself.

“Also, that there were two stones in silver bows—and these stones, fastened to a breastplate, constituted what is called the Urim and Thummim—deposited with the plates; and the possession and use of these stones were what constituted ‘seers’ in ancient or former times; and that God had prepared them for the purpose of translating the book (Pearl of Great Price/Joseph Smith-History 1:34-35).”

In the traditional sense of the word 'translation', there can be no agreement with the method described here, anywhere in antiquity. It is simply lost, in translation.

Latter-day Saints (LDS) believe:

History

The Book of Mormon was compiled by ancient prophets in the Americas on Gold plates and buried around 400 A.D. It is another testament of Christ, like the Bible, but from the American continent. Explanation of the Book of Mormon - lds.org

In 1823, a few years after being visited by God Himself, Joseph Smith was visited by the compiler of the Book of Mormon, a prophet from the ancient Americas named Moroni, who showed him in a vision where the plates were buried. He also charged him with the responsibility of translating the record from the ancient language in which it was written. Book of Mormon Translation - lds.org

Language

The book itself talks about the language it was written in. The first prophet, in the first book, Nephi, states in 1 Nephi 1:2:

2 Yea, I make a record in the language of my father, which consists of the learning of the Jews and the language of the Egyptians.

Then, 1000 years later, Moroni stated in Mormon 9:32:

32 And now, behold, we have written this record according to our knowledge, in the characters which are called among us the reformed Egyptian, being handed down and altered by us, according to our manner of speech.

33 And if our plates had been sufficiently large we should have written in Hebrew; but the Hebrew hath been altered by us also; and if we could have written in Hebrew, behold, ye would have had no imperfection in our record.

The document known as the Anthon Transcript is reported to be true, but there is controversy over whether or not the image online is actually of that original transcript. See Encyclopedia of Mormonism and More Insights on Anthon Transcript

Translation

Joseph Smith did not know how to read the text in it's original language. In fact the prophet Moroni said in Mormon 9:34

It's translation was done by the power of God using a seer instrument called the Urim and Thummim which also happens to be mentioned many times in the Old Testament (Eg. Exodus 28:30)

34 But the Lord knoweth the things which we have written, and also that none other people knoweth our language; and because that none other people knoweth our language, therefore he hath prepared means for the interpretation thereof.

Mormons believe that he translated the Golden Plates to the Book of Mormon, by using instruments such as the Urim and Thummim which were provided with the plates. The writing on these plates are of an unknown language, but are referred to as "reformed Egyptian".

To note, they don't believe it to be a translation of the Bible, but to instead be 'another testament of Jesus Christ'.

Joseph Smith reported that on the evening of September 21, 1823, while he prayed in the upper room of his parents’ small log home in Palmyra, New York, an angel who called himself Moroni appeared and told Joseph that “God had a work for [you] to do.” He informed Joseph that “there was a book deposited, written upon gold plates, giving an account of the former inhabitants of this continent, and the source from whence they sprang.” The book could be found in a hill not far from the Smith family farm. This was no ordinary history, for it contained “the fullness of the everlasting Gospel as delivered by the Savior.”

The angel charged Joseph Smith to translate the book from the ancient language in which it was written. The young man, however, had very little formal education and was incapable of writing a book on his own, let alone translating an ancient book written from an unknown language, known in the Book of Mormon as “reformed Egyptian”.

[...]

Joseph Smith and his scribes wrote of two instruments used in translating the Book of Mormon. According to witnesses of the translation, when Joseph looked into the instruments, the words of scripture appeared in English. One instrument, called in the Book of Mormon the “interpreters,” is better known to Latter-day Saints today as the “Urim and Thummim.” Joseph found the interpreters buried in the hill with the plates. Those who saw the interpreters described them as a clear pair of stones bound together with a metal rim. The Book of Mormon referred to this instrument, together with its breastplate, as a device “kept and preserved by the hand of the Lord” and “handed down from generation to generation, for the purpose of interpreting languages.”

- Book of Mormon Translation @ lds.org