When it comes to business decisions that benefit both, the B1G and SEC could absolutely sit down and make a deal over a playoff. When the dollar amount gets high enough, emotions are pushed aside and cold-hearted greed takes over. There are plenty of folks in the B1G and SEC that don't hate the other and they care a whole lot more about $$$ than anything else.
This is how big business works. I've seen it happen in my industry.