I hate to break it to the Dude, but AAU or not...if Oklahoma wants in the B1G, the B1G will take them. Their brand is too big for the B1G to ignore. They have the TVs, now they need the content. OSU vs OU would be a marquee matchup and add some much needed strength to the western part of the conference and a rival with Nebraska.
And adding Oklahoma to the schedule for the B1G's western schools would alleviate some of the angst that Iowa, Minnesota, etc. feel about the expanded conference and loosing annual games against the eastern marquee teams - Ohio St & Michigan. Adding Oklahoma would also put the B1G on the doorstep of Texas with respect to market penetration & recruiting.
The two issues with Oklahoma besides the XII and GOR challenges is that they 1) are not AAU, which I doubt is a requirement, is a consideration, and Oklahoma is not contiguous to the B1G (not taking Missouri when the B1G had a chance may be a major error on the B1G's part). That can be addressed by taking Kansas, too.
As for a partner, it would likely be between Kansas (Basketball, AAU, links the B1G to Oklahoma, Kansas City market) v. UConn (basketball, not AAU but academically higher than Kansas & other AAU universities, complements Rutger's footprint in the NYC market, entry into the Boston market, home market [CT] is larger than Kansas City & much more affluent, hockey, etc.).