Could be the shifter. Not on GT Sport, but it's been messing up on me in GT6. I've heard it's something that happens. I haven't seen anyone saying it's been faulty in GT Sport.
Fun fact: I've recently started playing GT5 again!
My shifter works fine with Assetto Corsa, Dirt Rally, and Project Cars. So I don't think it's my shifter having issues, but I do find it hilarious that my car starts furiously shaking when I shift into reverse!