Did הבתים transform into ויבן?

Summary: At first glance, it looks like Rashi miquoted a pasuk in Melachim. The Lubavitcher Rebbe explains the steps in its transformation, such that Rashi cited the pasuk correctly and scribes corrupted it. But perhaps Rashi never gave this pasuk as prooftext. I have a manuscript to back this idea up. Also, the prooftext is not very persuasive, since the midrash is speaking of the kingly and levite families, while the pasuk is speaking about physical structures, a palace and a Temple.

Post: I saw this interesting dvar Torah from the Lubavitcher Rebbe over Shabbos. Consider the following Rashi:

21. Now it took place when the midwives feared God, that He made houses for them.		כא. וַיְהִי כִּי יָרְאוּ הַמְיַלְּדֹת אֶת הָאֱלֹהִים וַיַּעַשׂ לָהֶם בָּתִּים:
He made houses for them: The houses of the priesthood, the Levitic family, and the royal family, which are called houses, as it is written: And he built the house of the Lord and the house of the king, (I Kings 9:1) [sic] 5, the priesthood and the Levitic family from Jochebed and the royal family from Miriam, as is stated in tractate Sotah (11b).		ויעש להם בתים: - בתי כהונה ולויה ומלכות שקרויין בתים. (ויבן) [הבתים] את בית ה' ואת בית המלך (מלכים א' ט א) (לקוטי שיחות חלק כ"א, ע' 2 הערה 29). כהונה ולויה מיוכבד, ומלכות ממרים, כדאיתא במסכת סוטה (סוטה יא ב):

The pasuk in sefer Melachim, 9:10, reads:

י  וַיְהִי, מִקְצֵה עֶשְׂרִים שָׁנָה, אֲשֶׁר-בָּנָה שְׁלֹמֹה, אֶת-שְׁנֵי הַבָּתִּים--אֶת-בֵּית ה, וְאֶת-בֵּית הַמֶּלֶךְ.

10 And it came to pass at the end of twenty years, wherein Solomon had built the two houses, the house of the LORD and the king's house--

so how could Rashi cite it as וַיִּבֶן אֶת-שְׁנֵי הַבָּתִּים--אֶת-בֵּית ה, וְאֶת-בֵּית הַמֶּלֶךְ? The Hebrew above refers to a sicha of the Lubavitcher Rebbe in which he explains that probably, initially Rashi cited the pasuk correctly. And so it said הַבָּתִּים. But then, Rashi would say the word בָּתִּים twice in a row. The second was written in short as הַבָּ'ש. Then, because heh sometimes looks in some manuscripts like a vav and a yud, some scribe reinterpreted it as a vav and yud, and interpreted the single quote at the end of the abbreviation to be an abbreviation for a nun sofit. Thus, habatim became vayiven.

This sounds possible.

Here are two other possibilities.

If we look at a ktav yad of Rashi from Rome, 1470, we don't see the offending misquotation of the pasuk. Instead, all we have is:

There is also no reference to specifically Masechet Sotah, as opposed to the gemara in general, or Masechet Megillah, which perhaps precedes, something which the Lubavitcher Rebbe also makes some diyukim about.

Now, earlier manuscripts from elsewhere do have the surrounding statements -- of בתי כהונה ולויה ומלכות שקרויין בתים. (ויבן) [הבתים] את בית ה' ואת בית המלך, and of כדאיתא במסכת סוטה. But I would suggest that these were the work of an overeager scribe, who inserted these 'helpful' comments into his Rashi text.

Indeed, the prooftext from sefer Melachim that the kingly and Levite family are called houses is not found in the gemara in Sotah. It offers other prooftexts to other aspects of the idea. Further, if we look at the pasuk in Melachim, it is talking about physical houses, namely the palace and the Bet Hamikdash. How in the world can it be a prooftext to the idea that The houses of the priesthood, the Levitic family, and the royal family, [...] are called houses? It seems like this was written by a talmid toeh, an erring student, rather than by Rashi himself. (A better proof would be, e.g. vayelech ish mibet Levi from elsewhere in Shemot, and maybe וְאֵת שֶׁבַע הַנְּעָרוֹת הָרְאֻיוֹת לָתֶת-לָהּ מִבֵּית הַמֶּלֶךְ from Esther perek 2.)

If so, it would not be so surprising if this erring student misquoted the pasuk in Melachim.

And it is not such a bad misquote, anyway. It gives the main gist of the pasuk, without being too verbose. Instead of the roundabout  אֲשֶׁר-בָּנָה שְׁלֹמֹה, אֶת-שְׁנֵי הַבָּתִּים--אֶת-בֵּית ה, וְאֶת-בֵּית הַמֶּלֶךְ, it communicates that someone built these houses. Say it is a minor rewording in place of an ellipses (...). Furthermore, look in the preceding perakim which discuss the building, and the word וַיִּבֶן  is indeed used. Thus, Melachim 6:1:

ט  וַיִּבֶן אֶת-הַבַּיִת, וַיְכַלֵּהוּ; וַיִּסְפֹּן אֶת-הַבַּיִת גֵּבִים, וּשְׂדֵרֹת בָּאֲרָזִים.

9 So he built the house, and finished it; and he covered in the house with planks of cedar over beams.

Chazal in the gemara often "misquote" pesukim. I've argued elsewhere that this was them following (and sometimes darshening) a Samaritan text. Other explanations are that the gemara, rather than the pasuk, was corrupted by scribal error (as was the Lubavitcher Rebbe's explanation of Rashi here). Another explanation is that they are deliberately misquoting the pasuk, for religious reasons, because of its status of Torah shebichtav.