No respect of persons does not come into play with fraternal correction?
I'm going to have to correct YOU on this one --
Fraternal correction has nothing to do with HUMAN RESPECT or RESPECT OF PERSONS.
That quote you're referencing "respecter of persons" is referring to GOD. Yes, to God we're all a bunch of ants. We are ALL beneath Him. The highest king and the lowliest homeless man. God will judge the highest king and the lowliest peasant, because He is "not a respecter of persons".
We, however, are subject to authority. An authority put over us BY GOD and we are commanded to respect the "persons" who are put above us. Not all men are equal, nor are they created equal. They are only equal IN THEIR IMMORTAL SOULS, BEFORE GOD. Men are clearly NOT equal on this earth: neither in wealth, nor talent, nor ability, nor status, nor power.
Having zero "human respect" is a goal for all Catholics.
Tearing down all human authority, rank, and position, as if we would pretend to emulate God, Who is above all -- that would be insubordinate nonsense.
With regards to "Fraternal Correction" just look at the words themselves: correcting a brother. That implies an EQUAL, not a SUPERIOR. It's not "Paternal Correction" or "Royal Correction" because (for starters) that simply wouldn't work 99,999 times out of 100,000.
Getting someone to listen to your fraternal correction is hard enough. Imagine correcting your superior! It's simply not your place. Pray for him, write him off, and move on. That would be the Catholic advice I believe. Can you produce something which contradicts this?