This fight between vaishnavite and saivite brahmins seems to be ridiculous. To me it appears that all brahmins were vaishnavites once upon a time. For example, can we start any religious ceremony without doing aachamanam and anga vandanam which is oblation to vishnu. Even in Sandhavandanam there is no prayer for shiva, though even navagrahas are prayed. The sandhyavandana mantras even say that all prayers go to vishnu. Likewise in all our rituals we recite mantras praising vishnu. My guess is that the puranas might have been written at that point of time when vaishnavism was followed by all brahmins and as a result show some bias against saivites. Subsequently when saivism was embraced by the kings, some of the brahmins must have become saivites for some reasons.
Even Paramacharya said that all brahmins were sporting only yaaga bhasma in the olden days. Wearing vibhuti and naamam came subsequently with the advent of sankara, ramanuja and madhva. (I would like to mention here that when I was living in Gujarat, I could not get vibuti even though the brahmins there are mostly saivites. I used to go to some temples and collect Bhasma from their for my use)
Hinduism is not an organized religion. Brahmins are not the only followers of Hinduism.
Sandhyavandana and other mantras has Vishnu's name and also Rama's. We also say 'Parameswara Priyadhartham'. What does Parameswara mean? Does it relate to Lord Shiva or Vishnu?
When Brahmin community underwent split, and Smartha sect was formed, who want to give prominence to Lord Shiva, the word 'Parameswara Priyadhartham' might have been included.
Regarding Bhasma, it is nothing but Ash. It can be in different form. So also using of Chandan. Chandan Abhishek, especially during Pradosham, is an important one in Shiva Temples.
Sporting Vibhudhi was prevalent even during Thevaram's time. Probably, you take only North or West India as your base for Hinduism.
It is not the fight between Saivite and Vaishnative Brahmins.
The point is some Saivite Brahmins have no guts to accept the fact some kings killed by Krishna were shown as demons and followers of Saivism, which, in my view, is not correct.
So many things have changed from olden days. Modern day religious leaders are travelling in AC cars.
Hence, Bhasma becomes Vibhudhi. But still we use Bhasma while performing Yagnas in the house.
Christians also give importance to Ash. Ash Wednesday is being celebrated every year.
Hence, Bhasma or Ash or Vibhudhi are one and the same, in different formulations.