Phew! After spending an hour chain fishing,I finally got your answer.
No,chain fishing does not give 3 perfect IVs past chain 60. I was chain fishing and I caught the Pokemon that came on 61st,62nd and 63rd chain( they were Clawitzers if you are wondering) and they did not have 3 perfect IVs.They had two pefect IVs.
Hope I helped! (And I really hope you like this answer because I spent a lot of time to find the answer to this question!)