Thankfully I don't have to register in TN, but it did confuse the guy at the FFL when he asked me what I was expecting and I answered "P-01". He said well, you might have the wrong gun then; so I had to explain it was the right one and why. Thankfully that was the end of it.
But I'm with armoredman, if you have to pick one, I would register it as a Compact D; if they come looking, they'll check out the slide, not the box...