There is a separate sect called Veerasaiva who worships only Shiva.
Why are you bringing all Vaishnavas here? Lot of NB Vaishnavas are having names of Saivite Gods, and are visiting Saivite Temples also.
The point raised here is about the attitude of Iyers following customs and conventions. As an Iyer, who wants to follow the customs scrupulously, I don't find any wrong in pointing out discrepancies. It is not the question of visiting only Shiva or Perumal Temple. It is proper adherence.
There are divisions among Iyers and Iyengars in customs and convention, naming children, even wearing Madisar after marriage etc. Such fact cannot be denied. A famous Iyengar cricketer, who has since retired, married an Iyer girl. His mother in an interview said that her daughter in law is from a different religion - Shiva Madham.
There is no harm in following one's religious practices without deviation and, at the same time, accommodating others. That is only true secularism.
If you want a united Brahmin community, different formalities should be abolished and create common code acceptable to all. I don't know why some Iyers want to degrade themselves in the name of unity. Unholy alliance will never succeed.
Instead of advising me, you can better tell Iyengar friends to change their practices, name their children with the name of Saivite Gods etc., and follow 'HARIYUM SIVANUM ONNU CONCEPT' seriously, and Jeers to stop temporary conversion.