Honestly, if all else truly is equal: i.e., you don't care about mileage earning, fare is the same, status isn't important, just book with the carrier operating the flight. Cut out the complexity. On day of operation, you'll be dealing with them anyway, and should anything happen before then, it's much easier to not have multiple parties pointing fingers at each other over who controls the ticket, who you need to deal with if you need a flight change, etc.