I do know that teams have to PAY to host games in the CBI and the CIT. It's actually a pretty steep fee. It's actually probably cheaper to play a road game in those tournaments than hosting one. I only know this because Navy was possibly going to be in one in 2009 (when we went 19-11) and the fee to host was astronomical, and there was NO WAY we could match the requirement for money made from the game.
In the history of the tournament, there has been ZERO schools from a power-six conference to be in the tournament, and was expanded to 32 teams this year for some reason. The only criteria is a winning record, since they obviously don't care much about facilities (hence American hosting) or anything else.