Is there any manuscript or historical evidence to support the final 13 verses of Genesis 50 that appear in the Joseph Smith Translation?

If you intend to exclude LDS specific sources, I await a more informed answer.

In the Book of Mormon a portion of those last verses are quoted directly. Here are links for comparison Genesis 50 JST and 2 Nephi 3.

From that it should be reasonably assumed that those writings were contained in the Plates of Brass (described here) as they contained "the five books of Moses," which would include Genesis.

As for the loss of the words, we have the following references as possible explanation:

And the angel spake unto me, saying: These last records, which thou hast seen among the Gentiles, shall establish the truth of the first, which are of the twelve apostles of the Lamb, and shall make known the plain and precious things which have been taken away from them; and shall make known to all kindreds, tongues, and people, that the Lamb of God is the Son of the Eternal Father, and the Savior of the world; and that all men must come unto him, or they cannot be saved.

1 Nephi 13:40

And also the word of God unto Moses:

And in a day when the children of men shall esteem my words as naught and take many of them from the book which thou shalt write, behold, I will raise up another like unto thee; and they shall be had again among the children of men—among as many as shall believe.

Moses 1:41

To start with, it might be helpful to clarify the terms used here.

Manuscript Evidence

Manuscript Evidence is a term that is well defined outside of Christianity, in the field of Textual Criticism.

Textual criticism (or lower criticism) is a branch of literary criticism that is concerned with the identification and removal of transcription errors in the texts of manuscripts. Ancient scribes made errors or alterations when copying manuscripts by hand. Given a manuscript copy, several or many copies, but not the original document, the textual critic seeks to reconstruct the original text (the archetype or autograph) as closely as possible. The same processes can be used to attempt to reconstruct intermediate editions, or recensions, of a document's transcription history. The ultimate objective of the textual critic's work is the production of a "critical edition" containing a text most closely approximating the original.

The basics are covered in another post on this site: What is “Manuscript Evidence” and how is it useful?, and it would be redundant cover the same ground here. The short version is this:

• The more copies we have of a text the more confident we can be that our current translations are true to the original text.
• Older copies are less likely to contain unintentional/copyist errors than newer ones, so the more old manuscripts we have, the better.

Historical Evidence

This would be external evidence such as works written by historians of the time, or close to the time, corroborated by accepted historical evidence - documents recording the events at the time, archaeology, etc.

Answer, now that the terms have been clarified:

When it comes to the final verses of genesis, we have *no strong manuscript evidence*. The evidence we do have is very weak, using the accepted forms of textual criticism. It is all recent, and all stems from the translation of one author, and a translation of original brass plates that are not available to study. We can only believe, with no external evidence, that they ever existed at all. All that we have at this time is the translated verses. We have no ancient copies of any texts that corroborate these verses. All we really have is the word of the founder, and other believers within.

When it comes to historical evidence, we also have very little. Before Joseph Smith, we have no historians referring to these verses, no Church sermons referring to them, nothing.

Of course, to the believer, lack of proof is not proof of lack. It's impossible to prove the non-existence of something. The answer to this question could, conceivably, be rendered null and void by the discovery of such manuscript or historical evidence. The lack of external evidence doesn't constitute proof that the verses are invalid. it just means that we have no corroborating evidence. All we have is the word of Joseph Smith and a handful of witnesses, whose trustworthiness is affirmed in the LDS Church, but questioned outside it.