Ron, I can't speak to what you have considered doing because I've never used, or purchased, T-track - not familiar with its properties.  I can only tell what I have done prior, found it to be successful without any problems arising.
First, since your doors are not frame and panel, I would suggest you do indeed add some form of anti-cupping support. I have done this on my projects when no frame was used, and again - no problems have ever come up. In truth, I have also used the following method on large panels in frame & panel applications as well.
My method is to use 3/4" by 3/4" battens, typically with the same wood used for the door, screwed through elongated holes one each end and one screw in the middle. Per the dimensions you provided, I would only use two battens, each installed about one quarter of the door height, from top and bottom.
I would assume you are going to finish the doors inside and out, so doing this should also help in keeping the door flat, as moisture into the door is eliminated.
I trim the battens, usually chamfering all around, for a more finished look.
That's about it for my thoughts, but again, you should do what makes you comfortable.  I'll be looking for this project post to see what you incorporate into your solution.
Any questions are welcome, Tom

PS: Don't forget, the big box stores sell small pieces of various hardwoods if needed.