it (especially the mid-major). I could see the conferences wanting the loser of its conference tournament championship game to receive the second bid. Most mid-majors would assume that their second bid would end up with a school from a power conference who they perceive to be an inferior team.

The public assumes that when the Committee selects the 34 at-large teams, then they are selecting the 34 "best" or "most deserving" teams. However, there is the possibility sometimes the Committee selects an at-large team for other reasons (i.e. majority of the Committee members are from a power conference so they are going to look out for their own, bias against a HC, bias against a school, etc). Also, if the regular season champion was in the conference tournament championship game, then what would prevent the conference from arranging so the regular season champion lost the game? Then there is the issue of a conference having co-regular season champions. I believe each conference has their own tiebreaker procedure for determining the seeds in the conference tournament. Sometimes though the seeding comes down to a coin flip or drawing numbers out of a hat. I don't believe any conference would be happy with using a coin flip or drawing numbers to determine who gets the 1 regular season bid. If anything, they would want all the co-champions to receive one.