1- The article says that the reason they sacrifice their hair is to express devotion to their Avoda Zara. Doesn’t that sound like the passuk:
ויאמר שמואל החפץ לה’ בעלות וזבחים כשמע בקול ה’ הנה שמע מזבח טוב להקשיב מחלב אילים
Look how the Rishonim explain the mitzva of Korbanos. The Rambam (Moreh 3 32, brought in Ramban Vayikra 1,9) says the main point of Korbonas is to remove tayva for Avoda Zara. The Ramban writes that the reason is to cause the Makriv to think that the behaima is in his place, and it is as if he was makriv himself.
If this is the case with korbanos that their are reasons given to explain that it is for the Makriv’s benefit, not to give the korban Kavoyachol to Hashem, than obviously tikrovos avoda zara is the same. If there is a maysa which is for the mussar purposes of the makriv it is mommish domeh to korbanos.
2- The teshuva from R. Y. soloveitchik (which wasn’t written lhalacha lmayasa as he writes) is not disputing any facts, but rather he is claiming that cutting hair isn’t similar to a maysa shechitah. Acharei Hamechila, it is ridiculous to claim that shechita, which is cutting simanim of an animal, is more similar to breaking a stick than it is similar to cutting hair. There are many more rayos if necessary.
3- In the article from Rabbi Yair hoffman he claims it is muttar because of a Sfeik Sfeika. Safek if the hair is given as an offering and a safek where the hair comes from.
That is straight out ignorance in the rules of Sfeik Sfeika for many reasons.
A- We don’t say sfeik sfeikos, if one safek is in the guf hadavar i.e. is it tikrovos or not, and one safek is because of a taaruvos i.e. if hair came from India. This is the most basic rule in Hilchos safek sfeika, see Yd 110, 9 and shach klalei sfeik sfeikos 1.
B- The Safek as far as what they believe, is a safek chisaron yideya which is not either mitzaref to safek sfeika. (see taz yd 98,6)
C- If there is a rov from India, which is pretty clear that that is the case, than most Achronim hold you do not say sfeik sfeika against a rov.
4- Tikrovos avoda zara has no bittul, this is muskam lakol, in the gemara (az 50) of the street paved from avnei markulis. the gemara says it is muttar to be nehneh from the street since it is not kein pnim.
there is a machlokes rishonim what that means, the raavad broght in the ritva says that tikrovos a”z sheino kein pnim is also assur, just it is possible to be mevatel. Most Rishonim hold that it is muttar and doesn’t need bittul.
Now, irrelevant to this, the Rambam rules that anything found inside the beis avoda zara is assur, and the BY and many learn that this includes things that are not kein pnim. Comes along the Bais Shlomo and says that the cases of tikrovos sheino kein pnim that the Rambam assurs, with a bittul the Rambam is maskim that it is muttar, just like the Raavad.
That has absolutely no Shaychus to our discussion. Kein pnim means either an object that was sacrificed bfnim, or a maysa domeh to pnim was done. I.E. Shchita assurs even a grasshopper acc. to the way the shulchan aruch paskens, since the maysa shechita is kein pnim. Same story with shviras makal, or any tolda of shechita.
Therefore in our discussion that the hair was cut, which is dumya to shechita that is kein pnim, the Beis Shlomo is totally irrelevant, and it is clear that there is no bittul.
5- the pri megadim that rabbi hoffman brings in oc 586 is not talking about tikrovas avoda zara, rather regular avoda zara, and vaiter has no shaychus.