What is the source for the Baal Shem Tov's signature?

It looks like I was wrong. The letter shown in Hatamim (on the page after the one referenced below) is a different one.

The above answer is flat-out wrong, for a very simple reason:

The letter in question isn't from the Kherson archive. Per information provided by the sixth Lubavitcher rebbe (here), this letter was purchased by one Yehuda Leib Zlatopolsky in 5613 (1853) and, some time later, gifted by his son Alter to the fifth Lubavitcher rebbe.

The topic is naturally considered controversial. The signature, as @Boaz wrote in the comments, comes from an epistle purportedly written by the Baal Shem Tov, discovered in what is now known as the "Kherson Geniza". The story of the geniza was related by Moshe Rosman in his book "Founder of Hasidism: A Quest for the Historical Ba'al Shem Tov", pp. 123-124:

"In 1916, a bookseller named Naftali Zvi Shapiro offered to sell a scholar, Abraham Kahana, letters written by the Baal Shem Tov that had been stored in some geniza (cache of old documents). By 1918, rumors about this cache were circulating throughout Jewish eastern Europe and appearing in the Jewish press. There was confusion as to where the geniza was, in St. Petersburg, Kiev, or Kharkov. Finally, in 1921, the letters appeared in Odessa, having originated (supposedly) in the government archive in Kherson. The story was that in the confusion following the revolution in 1917, the letters were removed from the Kherson archive and brought to Odessa.¹ There most of the letters were purchased by the hasid Shmuel ben Shneur Zalman Gurary and given to him to the leader of Habad Hasidim at the time, Rabbi Shalom Dovber. After piecemeal publishing efforts, in the 1930's his successor, Rabbi Yosef Yitzhak, published some three hundred letters, purportedly from the pen of the Besht or his family members and associates, in the Habad journal HaTamim. The discovery of the letters caused an initial rush of excitement among hasidim and academic scholars alike. It soon became apparent, however that the excitement was misplaced. Close inspection of the contents, form, and paper cast serious doubt on their genuineness, and today, outside of the Habad Hasidic movement, there are virtually no authorities who consider these letters anything other than forgeries."

The points for the case of forgeries he makes are as follows:

• The sellers of the items claim they had originally been kept in the home of Rabbi Yisrael of Ruzhin and were confiscated from when he was arrested circa 1838. While each letter was found sealed with a stamp indicating it was found in the house of the rabbi, the scenario is strange because until the discovery, there was no Ruzhin-based tradition that the Rabbi of Ruzhin had such an archive.

• Furthermore, no one saw the letters in Kherson. The identity of the "liberators" was never revealed. Why were they even transferred to Kherson? Why did Shapiro offer to sell them to Kahana as early as 1916, if they were liberated during the Revolution?

• The Russian on the stamps is not idiomatic and seems to come from someone whose mother tongue was Yiddish.

• The Yiddish itself is not Podolian in style and the Hebrew contains several modernisms.

• The language style is uniform, despite the letters in total having purportedly been written by about thirty different people.

• The dates are all given in the same manner - the Hebrew year + the Torah portion of the week + the day of the week. Cross-referencing with calendars, it turned out that some of the letters were written on days in which letter-writing is not allowed for Jews, such as the last day of Passover, Shemini Atzeret and the 9th of Av.

• Various other anachronisms and inaccuracies. For example, the Maggid of Mezeritch, known to have been from Mezeritch, is said to have come to the Baal Shemt Tov from there, when in reality he met him coming from Lukach.

• A sample of one of the papers was tested at a technical institute in Vienna at the behest of Abraham Schwadron. The scientists established that the paper on which the letters had been written could not have been manufactured prior to 1846, eight years after the Rabbi of Ruzhin was arrested. Schwadron's research was published in an article in the weekly publication מאזניים (Mozna'im) and can be viewed here.

In a footnote on this last point, Rosman wrote that defenders of the letters state that it's true that these aren't the original letters, but they are copies of authentic letters, and on occasion, mistakes crept in.

Interestingly, not all members of Chabad defend the Geniza. Rabbi David Tzvi Hillman published many Chassidic letters and epistles in his book "אגרות בעל התניא ובני דורו" (The Epistles of Baal Ha'Tanya and His Generation) and dedicated a section at the end of the book to the Kherson Geniza. In his view, the Geniza texts are forgeries. I recommend checking out the book for a more expansive list of examples of anachronisms and other such mistakes than what Rosman brought.

Dr. Yitzchak Rafael, one of Rosman's sources, wrote an essay on this controversy, "גניזת חרסון" (The Kherson Geniza), Sinai 81. There he wrote that some of the problematic aspects of the texts may actually be used to prove their authenticity, and thus we may be able to understand why the Rabbis of Chabad believed the letters were authentic:

• The mistakes prove that they weren't created by a forger who wanted to make a buck. They were copied from originals written by people with deep Kabbalistic knowledge.

• The letters might have been copied in a hurry, on the command of the Rabbi of Ruzhin, explaining some of the mistakes.

Rafael himself concluded that the middle approach seems to him the most likely: The texts are copies and were made in a hurry at the behest of the Rabbi of Ruzhin and this is what caused the mistakes. Most of the copies are to be accepted as based on authentic letters, though it's possible that actual forgeries made there way into the collection.

It should be noted that the Lubavitcher Rebbe himself also addressed the controversy and brought his own list of reasons as to why the letters are to be deemed, at the very least, copies of authentic letters. This can be viewed in the journal "הספר" (The Book) in two parts, one in the first issue and the other in the second issue. In the latter issue, Rabbi Hillman also replied to the Rebbe.

The long and short of all of this seems to be that even if the content of the letters is authentic, the texts themselves are copied, as well as the signatures, which would mean that we cannot be entirely sure that the signature of the Baal Shem Tov as depicted there indeed looked exactly like that.

¹ It should be noted that in Rafael's essay on the subject he wrote (pp. 132-134) that there are many different versions of the story, not fully in agreement with one another.