The short version is no, DTM does not expose the rule name for you to reference.
I have spent a fair amount of time trying to find some unofficial method or property with it, but it is simply not exposed within the scope of anything you can passively do.
I have also spent a fair amount of time figuring out the least invasive way to alter the _satellite object to give me what I want, and.. while I have done it in a sandbox, it is not something I have or will ever actually use in practice. (Adobe won't even support passively using existing internal methods/properties, let alone actually changing the DTM library).
The good news is this is something a lot of people have asked for over the years, and Adobe has listened and added it to Launch, if/when you migrate from DTM to Launch.