It is noted that people born under certain Nakshatras (stars) must perform Shanti Parikaram (remedial rituals) during a lunar eclipse. Do those who have performed Bharanyasam (total surrender to God) to Perumal also need to perform these remedial rituals? If so, how should it be done? 

There is no custom for those who have performed Bharanyasam to do a separate Shanti Parikaram during the eclipse period. During that time, they light lamps at their shrine at home or in temples. Activities like Pradakshinam (circumambulation), reciting Stotras (hymns), and Japa (chanting) can be done. Otherwise, there is no necessity to perform Shanti Parikaram in the form of a Sankalpam (formal ritual vow).