Thank you for bringing this to my attention. I have looked into the matter, you are correct that it is certainly not the ideal time to say them (S.A OC 581:1 and mishna berura there) but under extenuating circumstances Rav Moshe (OC 105) permits it because technically there is no prohibition but shouldnt be done yearly and each time needs a heter. Some say since in eretz yisrael its the correct time and is an ais ratzon there, that is enough to permit if one has work early next day. A person I asked told me that it is brought in halichos shlomo footnotes in moadim that one should not say the 13 middos with a minyan before chatzos (kabbalistic reasons) but others because of above reasoning permit it.
I did not want to get into the nitty griity's. I was just trying to make an awareness to an apparently not so well known halacha.
That Reb Moshe you are referring to is very strongly apposed to saying selichos before Chtatzos. He also says that if it is neeeds to be done, it should be announced that it is a Horaas Sha'ah.
At the end of my comment, that Reb Moshe zt"l is what I had in mind.
In Halichos Shlomo, if I remember correctly, he was also extremely against this. But in the footnote you rederenced he says he found out there were some kehilos in Poland who already had this minhag, so he laid back a little bit.
The Mateh Efraim (where most of the Mishna Berura on these halachos are sourced from) says that if one finds himself in a Minyan saying selichos before Chatzos, he should sit there and not say along or even answer the yud gimmel middos. He says that on a regular night one may say the "viduy" part of selichos, but NOT on Motzei Shabbos. This is the only instance I am aware of where one advised to be "presh min hatzibur".
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