Here's a link for reference:

http://en.wikipedia.org/wiki/College_Football_Playoff


But basically, in years when the championship bowls are not part of the 4-team playoff, they get the highest ranked non-playoff teams from the following conferences:

"Contract Bowls":

Rose Bowl: Big Ten vs. PAC-12
Sugar Bowl: SEC vs. Big 12
Orange Bowl: ACC vs. SEC/Big Ten/Notre Dame*

"Access Bowls":
Fiesta Bowl: at-large or "group of five"
Cotton Bowl: at-large or "group of five"
Atlanta Bowl: at-large or "group of five"

*Orange Bowl takes the highest between the #2 ACC team, #2 Big Ten Team, and Notre Dame

The highest ranked champion of the "Group of Five" is GUARANTEED a spot in one of the bowls if they aren't in the playoff.
The remaining at-large bids will be decided by the selection committee


So the only way that BYU would be able to get any sort of bid to the Cotton Bowl would be to

1. Be the highest ranking conference champion from the MWC/AAC/CUSA/MAC/Sun Belt. This is impossible without joining one of those conferences.
2. Be ranked high enough to warrant an at-large selection into the bowl game (essential a top 12 finish in the selection committee's rankings).

There is NO WAY that BYU will be able to contract with the Cotton Bowl because the Cotton bowl is considered one of the "access bowls" which are by definition not supposed to have any auto-bid tie-ins (thus creating "access" for at-large teams). The Cotton Bowl in years that it's not a playoff bowl is basically in the same place as the non BCS Championship BCS games were in that only the top teams are eligible to play in it.