When you spoke with your child's mother about why she wants to use the PT in Murfreesboro, what happened? You wrote that you'll be taking the kid half the time, which means the mother will be taking him the other half of the time. If Murfreesboro is the midway point, why should she have to do all the driving? You made much of being the "primary custodian," but if each of you is handling half of the appointments, it doesn't matter that you're the "primary custodian." If you're each handling half of the visits, why should all of the travel burden fall on the mother?
You don't need to answer that, but you should prepare yourself to explain it to the court. Here's the reality. If she wants to take the kid to a PT in Murfreesboro on her custodial time, there's nothing stopping her from doing so unless you seek relief from the court. Likewise, if you want to take the kid to a PT in Nashville on your custodial time, there's nothing stopping you from doing so unless she seeks relief from the court. Are you going to do that? Is she going to do that? The court is going to expect that you'll behave like adults and is not going to want to micro-manage your lives.
You don't need to answer that, but you should prepare yourself to explain it to the court. Here's the reality. If she wants to take the kid to a PT in Murfreesboro on her custodial time, there's nothing stopping her from doing so unless you seek relief from the court. Likewise, if you want to take the kid to a PT in Nashville on your custodial time, there's nothing stopping you from doing so unless she seeks relief from the court. Are you going to do that? Is she going to do that? The court is going to expect that you'll behave like adults and is not going to want to micro-manage your lives.