It seems that "Shamshon" is simply how the name was pronounced in Lithuania, and indeed, Russia. For example, we find the historian Jacob Schamschon of Kovno:
One might surmise that the Lithuanian tendency to exchange /sh/ and /ss/ sounds was at work. (Most people know that Lithuanians did not pronounce the shin. Less well known is their tendency to change a sin to a shin. Sibboleth, indeed.) So "Shamshon" was simply the pronunciation of "Samson" in that particular Yiddish dialect. Do not be surprised at the idea of a "Yiddish" name being based on the Gentile version of the name (Zalman = Salomon). Here's a page from a subscription list from an 1818 edition of the Five Megillot:
And these are only some of the Moses. (Or is that Moseses?) The next page has 17 more.
An extremely useful book, published in Warsaw, 1908, makes it clear that this is the pronunciation in what it calls Russia (Rusland):
And note that it also transliterates it Schimschojn, the Polish (or shall we say Pojlish) pronunciation. (See יודישע שפריכווערטער און רעדענסארטען by Ignatz Bernstein and B. W. Segel)
So we've solved that element. If anyone wants to know why yeshivishe people say "Shamshon Refoel," it's because that's what the man was called in the heartland of where yeshiva culture developed.
But it's not as simple as that. Although I surmise that Rabbi Hirsch actually went by Samson - pronounced something like Zamzun, but with the "u" being more like a combination of the English "u" and "o" - there are also many sources indicating that "Schamschon" was a German pronunciation as well.
Before I get to these, let's go back to the beginning. Although the Masoretic text of the Bible spells it with a chirik, and thus "Shimshon," the vocalization in Tanakh is late. So although an ounce of tradition is worth a ton of acumen, and we are not to disregard this, the Septuagint is an earlier vocalized source, and it uses an alpha - Σαμψών, not Σιμψών. This is why the name is "Samson" in English and the various other tongues. Interestingly, with the rise of Protestantism and an interest in the Hebrew Bible, the alternate form Simson, based on the Masoretic text, entered the Christian consciousness. See, e.g., Luther's Bible, where he writes "Simson." I guess we have to thank Martin Luther for the Simpsons, whom otherwise might be the Samsons, or more likely, the Smiths.
In any case, there are a number of (Jewish) German sources which use "Schamschon." For example, this excerpt from an 1887 article in the Israelit refers to Rabbi Schamschon Ostropolle:
Lest it be argued that this was an attempt to give his name in "Jargon," as he was a 17th century Polish rabbi, here is an anecdote about the 17th century Viennese Court Jew Rabbi Samson Wertheimer, where he is called "Rabbi Schamschon Wien," evidently his 'traditional' name:
Thus, it seems to me that there was at least the perception that this was the traditional way in which German Jews pronounced the name, no less so than in Rusland. In reality, probably many German Jews pronounced it that way still in 1905, which is when the excerpt above appeared. Since it is a great anecdote, I will explain what it says.
Rabbi Samson Wertheimer employed a tutor for his family named David, who was an apt Talmudist. He later received a position in the rabbinate, on R. Samson's recommendation. Since he was just the tutor working for him, he couldn't bear to call him "Rabbi," but continued to call him David, as before. Once R. Samson was on a trip, and stayed in the community where David was. He visited him, and saw that Rabbi David was studying Mishnah Shabbat 3:12, where it says "If he wrote [two letters; the name] 'Shem' (shin-mem) of 'Shimon' or of 'Shemuel'." So he asked Rabbi David, why didn't Rabbi, the Redactor of the Mishnah (and in fact the direct source of this statement!) say "Shimon, Shmuel, or Shimshon"?David answered, "The reasons seems to be that Rabbi did not say "Shimshon," because Shimshon did not say "Rabbi!"
Here is what "Reb Shamshon Wiener" looked like, by the way. From Wikipedia.
Hopefully we have proved our case, that "Shamshon" is simply what the Jews over a wide geographic area actually called their Shimshons.
Finally, I came across the following in a list of Jewish folk expressions, from 1898.
Schimschon in Ruh und Schamschon in der Wickel.
Something like "Shimshon in calm, and Shamshon in distress." Does anyone have any idea what this is supposed to mean? If I had to guess, I would say that it is like a reference to correctness, Shimshon, and the vulgar jargon, Shamshon. When calm, he is a sedate gentleman, and he says Shimshon. Rile him up, and he reverts to the ghetto Jew, Shamshon.
 Of course, strictly speaking this is not true. There are many people today who are named for him, or named for those named for him, who also use the form "Shamshon."