I have reached the point now in my first CL playthrough where Caesar has asked me to destroy the Brotherhood of Steel. I agreed to do it but... the BoS is my favorite faction throughout the series and destroying a chapter of them is not something I want to do. He did say when pressed that if he could be convinced that the benefits of an alliance with the brotherhood would outweigh the risks he would consider it. If I complete the quests for the brotherhood will I be able to convince them to ally with Caesar and convince Caesar to accept it?

I haven't done a Legion run but I doubt they would. Your best bet is to do a special run for Legion and House and in that run don't make friends with the Brotherhood so it won't be a betrayal when you whack them to advance the story. That's what I'm doing.

Well Elder McNamara can be convinced to ally with the NCR so perhaps in theory Hardin can be convinced to ally with the legion. But I doubt it.