Amazingly, there is one source on Otzar Hachochmah that refers to both the Tooth Fairy and a Jewish alternative: The Candy-throwing Angel. In the book ופקדת נווך by R' Erez Abrahamov, p. 70, it says (emphasis in the original):
"וראיתי בספר "חנוך לנער" המבוסס על דבריו של הגאון הרב חיים קנייבסקי שליט"א (עמ' ל"ז ואילך) שדן בכמה נקודות מעניינות לחינוך לשקר או מידות רעות:
"פיית השיניים" והמלאך זורק הסוכריות - בספר "בטוב ירושלים" לרב ידלר מובא מנהג ירושלים, שהיו מביאים את הילד לראשונה לחדר ללמוד, אזי היו זורקים עליו ממתקים ומעות מלמעלה ואומרים לו שהמלאך זורק לו את זה מהשמיים ואם הוא ילמד הוא יקבל מהמלאך הרבה יותר, ושאלו את הגאון הרב קנייבסקי שליט"א כיצד הדבר מותר, וענה:
"...אומרים כן בעלמא והילד יודע בסופו של דבר שלא המלאך הוא הזורק. אנחנו תמיד ידענו שזה לא נכון, ואפילו אם הילד לא יודע - הוא יידע"."
"And I saw in the book "Chanoch La'naar" that is based on the words of the gaon Rabbi Chaim kanievsky (p. 37 and onwards) discussing a number of interesting points regarding teaching to lie or bad middot (traits):
"The Tooth Fairy"1 and the Candy-throwing Angel - in the book "Betuv Yerushalayim" by Rabbi Yadler a custom of Yerushalayim is brought, where they would bring a child for the first time to the Cheder to learn, and would then throw upon him candies and coins from above and would tell him that the angel threw it on him from the heavens and if he will learn, he'll recieve from the angel much more than that,2 and they asked the gaon Rabbi Chaim Kanievsky Shlit"a (now Zt"l obviously...) how such a thing is allowed and he answered:
"...they say so as a common form of exaggeration and the child eventually comes to know that it is not the angel that throws. We always knew that this is not true, and even if the child does not know - he will know."
On the minhag of the Candy-throwing Angel there are more sources.
First of all, Rabbi Shmuel Halevi Horovitz in his book שבח הברית, p. 142 sources this custom to a book called ספר בראנדשפיגל, which, from what I gather, is more commonly known as ברנט שפיגל (for more info on the book, see here and here). However, it's in Yiddish and I have not been able to find any related keywords from a quick search. Rabbi Horovitz wrote that this book sources the custom to the book שבילי אמונה (one of several editions), but there I have also been unsuccessful in finding the relevant source in the book itself. With that said, the relevant source from Shvilei Emunah is brought in R' Simchah Asaf's book לתולדות החינוך בישראל, pt. 1, p. 59. I suspect that this is where other sources got their quote from.3 In this source, the angel is nameless and instead of candy, fruit were tossed.
Second, a discussion on why this minhag is okay by the Rebbe of Chabad can be found in Torat Menachem 5720, pt. 1, pp. 82-85. In the Rebbe's version, the angel is said to be Michael.
Side-question: Going back to the first source that connects the angel to the fairy, I am not entirely sure why the comparison should hold, given that the angel, among other things, represents a kind of "מתוך שלא לשמה יבוא לשמה", whilst what positivity does the fairy bring about exactly?
1 In terms of the format of the quote in the book ופקדת נווך it looks like the Tooth Fairy is explicitly mentioned in the book חנוך לנער, but it's not. This appears to be a connection made by R' Abrahamov.
2 הרב בן ציון יאדלר, בטוב ירושלים, עמ' נ' - Rabbi Ben Tzion Yadler, Betuv Yerushalaim, p. 50
3 Interestingly, R' Yisrael Shtzifanski in התקנות בישראל, vol. 4, p. 331-332, footnote 496 argues that since the Brantshpigl brought this custom as well as a number of others in the name of Shvilei Emunah, that means that this was no longer a custom in his time and/or place, otherwise he would have simply stated it was a traditional custom.