I'm aware that Rov Moshe ruled that cholov Stam is okay under pressing circumstances. Is there leniency in consuming cholov stam if one can easily afford Yisroel, but wants to save some money? Or if there's a stam product that tastes much better, or doesn't get moldy in 2 minutes like cholov Yisroel? Conversly, which sources prohibit this lenient use? Thanks.
First, you are mistaken about Rav Moshe's position. I know from his son Rav Reuvein that they routinely had chalav hacompanies at home. Rav Moshe held chalav yisrael was no required when the milk had USDA certification. Full stop. He did say it was a nice chumerah (stringency) for schools, as a teaching opportunity. Now, on to your answer.
Rav Moshe coined the term "chalav hacompanies" to distinguish it from the prohibited milk, "chalav stam" (literally: just plain milk). But most observant Jews in the US were drinking it already before Rav Moshe got to America, never mind writing any of his series of teshuvos on the subject. And this was not a lack of observance, but the ruling of rabbis like R Jacob Joseph (rav of New York), the Ridbaz (R Yaakov Dovid Wilovsky, when he was rav of Chicago), R Yisrael Avraham Abba Krieger (rav of Boston), and others.
According to the Chazon Ish (Yoreh Dei'ah 41:4), the precedent was 17th century Italy, which also had government regulations on milk. The Peri Chadash (R' Chizkiya da Silva) ruled (Yoreh Dei'ah 115:6) that chalav yisrael was a ruling in the gemara. Not a new law, but an application of existing halakhah to the fact that we cannot really know what's in the milk we buy. And since they did have reason to believe the cow's milk was really entirely cow's milk, the gemara's ruling didn't apply.
The Chasam Sofer (YD 107) disagreed, saying that chalav yisrael was indeed its own derabbanan (a piece of rabbinic legislation) and is therefore prohibited whether or not the fear of non-kosher milk is realistic. And this is the position of the Pischei Teshuvah (YD 115:2 cites the Chasam Sofer), the Chayei Adam (Chokmas Adam 67:1) and the Arukh haShulchan (YD 115).
Rav Moshe Feinstein (starting with Igeros Moshe YD vol 1, #46-48) had to defend existing norms of his new community. He didn't use the Peri Chadash; rather he argued that even according to the Chasam Sofer chalav hacompanies is permitted. Rav Moshe gives other examples where knowing with some certainty is equivalent to seeing. And therefore chalav hacompanies is "watched" and therefore conforms to the legislation.
In that sense, Americans only think they're following Rav Moshe and his take on the Chasam Sofer when they drink chalav hacompanies. This is only because his teshuvos were the first in print.
And notice that either way, drinking chalav hacompanies isn't framed as a qulah (leniency), but baseline law. As per Rav Moshe's own consumption, and his phrasing in all his teshuvos on the subject, except for the one (YD vol II, #35) written to a day school adminstrator, where he makes an argument about teaching the idea of willingness to sacrifice to do beyond the minimum.
When can you be lenient? How much motive? Rav Moshe would actually allow you (if you're not shopping for a yeshiva) to simply buy whichever you choose. Maybe he would say -- although I am just guessing -- it is better to buy chalav yisrael because it means supporting more Jews.
To be clear about my own biases: R Kreiger, the aforementioned rav of Boston who was among those I cited who allowed chalav hacompanies, was my great-grandfather. I personally grew up drinking chh"c, and my family still does. I personally switched to chalav yisrael only when my first cycle of Arukh haShulchan Yomi got to Yoreh Dei'ah 115. (Compounded by the fact that so much of the junk I shouldn't be eating but was buying from the vending machine at work anyway were OUD.)