If non-stop flights out of DTW to your destination are important to you, you probably should switch and fly solely DL, and acquire status there.

How likely is it that you and your team can do productive work during a 4 hour trip DTW-BNA? It generally involves the nature of the work and the personalities involved. I know that I would feel pretty good about my ability to get work done if alone, not confident at all if I had others with me that they would work (and less confident that I would). If you don't think work would be done, you have some 'splaining to do in one direction or the other:
1. If you expect your employees to get a full day's work done that day in addition to traveling, you have to explain to them why you have extended their work day by 3 hours to suit your desire to get Platinum status on AA for yourself.
2. If you don't expect your employees to get a full day's work done, you may have to explain to someone higher in your organization why you have reduced the amount of work you expect out of them to suit your desire to get Platinum status on AA for yourself. In addition, you are sending a signal to your employees that personal goals are more important than work.