1) I found one of the Mefarshim who clearly understands that it is exactly 3,000. This is the Iyun Yakov printed in the Ein Yakov. He writes that the reason it is 3,000 is that they mourned for Moshe Rabeinu for 30 days. Now the question is: how many Halachos should one learn every day? The Iyun Yakov argues that in the same way that one should say 100 Berachos every day, so one should learn 100 Halachos every day. Since both Berachos and Halachos lead you to the fear of Heaven, it is logical that they should be equal. That means that in normal times they should have learned 3,000 in the space of 30 days, and since they did not learn because of the mourning for Moshe, this is why they forgot all those Halachos.

2) However, the Chida, in Birkei Yosef (Orach Chaim 155:11), disagrees with this. He writes that it is too much to expect everyone to learn 100 Halachos every day. He argues that since we say, "Kol ha;Shoneh Halachos b'Chol Yom Muvtach Lo she'Hu Ben Olam ha'Ba" -- "Everyone who learns Halachos every day will merit the Next World," it follows that "Miyut Rabim Shenayim." In other words, Chazal tell us that someone who learns Halachos every day will go to Olam ha'Ba, but they do not tell us explicitly how many Halachos. We therefore say that the minimum of the plural word "Halachos" is two. This is the conclusion of the Birkei Yosef and other Poskim who rule that, according to the letter of the law, one can merit the Next World also with learning two Halachos every day (see Sha'arei Teshuvah, printed together with the Mishnah Berurah in the standard editions, to Orach Chaim 155:1).

3) We still do not have an answer for why it is 3,000, but I am now going to break off and answer your question, do we find 3,000 used in other instances to represent a large number? I suspect you have in mind the Gemara in Chulin 90b, where Shmuel said that we find in three places that the Chachamim spoke with "Lashon Havai" -- exaggeration. The Mefarshim write that Shmuel is referring to the Mishnah, while in the Gemara there are many more examples.

4) However most of these examples seem to be with the number 300, not 3,000 (see Toras Chaim there in Chulin). The Rashbam to Pesachim 119a (DH Masuy) writes that whenever the number 300 is mentioned in Shas, it is not exact. I saw a beautiful explanation, cited by the Mesivta edition to Chulin 90b, in the name of the Chosen Yeshu'os, for why it is always the number 300. We know that the Hebrew letters of the word "Emes," truth, all have a solid base, while the letters of the word "Sheker," falsehood, all have a shaky base (see Shabbos 104a). This is because falsehood has no legs on which to stand, and very soon it will totally collapse. The letter "Shin" in a Sefer Torah stands on one thin point, where its thre branches come together. We see that the letter Shin represents falsehood. Since the Gematriya of Shin is 300, whenever Chazal wanted to indicate that the number they are giving is not serious, they used the number 300.

5) Back to your original question.... The truth is that I do not yet have an answer for how Chazal knew that it was 3,000, but it seems to me that there is no reason to say that the number 3,000 is an exaggeration. What I mean is that it is possible that it was not exactly 3,000, but it was not far off. It is not the same as "Lashon Havai," mentioned above (3), which was a deliberate exaggeration. We do find sometimes in the Gemara (see Sukah 8a) the concept of "Lo Dak" -- a number is used which is only intended to be approximate. That might also be the case with the 3,000 Halachos, but the real figure was not far from 3,000.

6) Presumably, Chazal possessed a tradition about how many Halachos were forgotten. We see that they also knew how to break this figure down: 1,700 Kal va'Chomers, etc. There is no indication that this is an exaggeration. This is different from Chulin 90b, where the Gemara says that the number of 300 Kohanim given, who were required to lift up the "gold vine" in the Beis ha'Mikdash, was a deliberate exaggeration (see Insights to the Daf there).

Kol Tuv,

Dovid Bloom

I posed your question to a Gadol and he replied that the number 3,000 used here, is precise.

Dovid Bloom

I asked the Gadol how Shmuel, who made the statement in our Gemara that 3,000 Halochos were forgotten, knew that it was exactly that number.

He answered "Kach Kibel" -- this is what Shmuel received as a tradition.

It must be that the Amora Shmuel heard from his teachers as a Kabalah that it was precisely 3,000 Halochos that were forgotten.

Gmar Chasimah Tovah,

Dovid Bloom