Reb Boruch, this is a very strong question.

1. I will strengthen the question a little by citing the Ramban to Devarim 6:18 ("v'Asisa ha'Yashar veha'Tov"), who writes: "And our Sages have a beautiful Midrash on this, for they said, 'This is Pesharah and Lifnim mi'Shuras ha'Din.'" In other words, the Ramban cites Rashi's explanation on "v'Asisa..." but he seems to say that the Mekor for this is from "Raboseinu," i.e. from Chazal. I looked up four contemporary editions of the Ramban on Chumash with annotations and they all note that the source of the Midrash that the Ramban cites is unknown. Hence, the commentators on the Ramban have already noticed that there is a problem with this way of explaining the verse of "v'Asisa."

2. I do not know if I have actually found a source for Rashi, but b'Ezer Hash-m I will try to explain this Inyan so that we can understand a bit about what is going on. My main source is the Maharsha on the Gemara in Avodah Zarah (25a), which you cited. The Maharsha there (DH Zeh Sefer Mishneh Torah) asks why should Sefer Devarim be called the "Sefer ha'Yashar" just because the word "Yashar" is written there once. The Maharsha answers that the reason is that whenever the Gemara cites the verse of "v'Asisa" it does so in connection with the concept of "Lifnim mi'Shuras ha'Din." He cites the Gemara in Bava Metzia (16b) that says that "Shum Hadar" -- if the Beis Din gave the lender the field of the borrower because he did not pay back his loan, and afterwards he pays back, then the borrower gets his field back. The Gemara derives this law from "v'Asisa." However, the Maharsha points out, the actual reason for why the field is returned is because of "Lifnim mi'Shuras ha'Din." What this means is that "v'Asisa" and "Lifnim mi'Shuras ha'Din" are really one and the same thing. According to strict logic, the borrower does not deserve his field back, but the Torah goes slightly beyond strict logic and tries to help people, and gives him back his favorite field. The Maharsha writes that the same thing applies to the law of Bar Metzra, that the neighbor has first right to buy the adjacent field. Logically, he is no better than anyone else, but the Torah does more than what logic requires and makes life easier for the neighbor by giving him the opportunity to buy the field. This Halachah is derived from "v'Asisa" in Bava Metzia 108a, but the Maharsha understands that it is another example of "Lifnim mi'Shuras ha'Din" because the two concepts are really one and the same. The Maharsha writes that after the verse in Devarim 6:17 instructs us to fulfill the Mitzvos and Edus and Chukim, it instructs us "to do the upright and good thing in the eyes of Hash-m." Even though Hash-m did not command us explicitly about a certain thing, we should still always look for what is the good thing to do.

3. I have another source to prove that Rashi learns that these different verses -- which instruct us to behave nicely to others even though there might not be a specific Halachah about it -- are all saying the same basic thing. The source is the Gemara in Bava Metzia (83a), which you also cited, which relates that the hungry workers broke the barrels but the employer was told that he nevertheless must pay their wages. The verse, "Lema'an Telech b'Derech Tovim," is cited, but Rashi (DH b'Derech) writes that this means "Lifnim mi'Shuras ha'Din." Tosfos earlier in Bava Metzia (24b, DH Lifnim) writes that "Lifnim mi'Shuras ha'Din" does not go that far; one would not be required by "Lifnim mi'Shuras ha'Din" to pay workers who had caused him a great loss. That is why the Gemara needs to find a different verse. However, Rashi does not learn like Tosfos, because Rashi udnerstands that "Lifnim mi'Shuras ha'Din" is a general Mitzvah which instructs us to go beyond the minimum of what the Halachah expects from us. This is what "v'Asisa" also teaches. Since the Torah uses a number of different verses to teach that one should go beyond the minimum required of him, Rashi in Va'eschanan puts "v'Asisa" together with the concept of "Lifnim mi'Shuras ha'Din" because it is all one Inyan.

Kesivah v'Chasimah Tovah,

Dovid Bloom

Here is a different source for the words of Rashi to Devarim 6:18. The source is the Sifri to Devarim 12:28. The verse there states, "When you will do the Tov and the Yashar in the eyes of Hash-m...." The Sifri (printed in the Malbim editions of the Chumash) cites a dispute between Rebbi Akiva and Rebbi Yishmael. Rebbi Akiva maintains that "Tov" means what is good in the eyes of Shamayim, while "Yashar" refers to what is upright in the eyes of man. Rebbi Yishmael disagrees and explains the opposite way: "Yashar" means in the eyes of Shamayim, while "Tov" means in the eyes of people. (Rashi ti Devarim 12:28 cites Rebbi Akiva's opinion.)

According to both opinions, one must do what is good in the eyes of Heaven. This implies Lifnim mi'Shuras ha'Din, because one who does something in this world that Beis Din cannot force to do causes happiness in the eyes of Shamayim. This is why Rashi (Devarim 6:18) learns that "ha'Yashar" and "ha'Tov" must include Lifnim mi'Shuras ha'Din according to everyone.

Rashi preferred, in his commentary on the Chumash, to cite the Sifri rather than the Gemara in Bava Metzia (30b), because the Sifri is a commentary written by Chazal on the Chumash, and thus it is a more appropriate source to cite to understand the meaning of the words in the Chumash.

Kesivah v'Chasimah Tovah,

Dovid Bloom