Just out of interest how close could you get 200mm lens possible?
I just checked. In descending order 90mm, 85mm, 125mm. So, you'll be fine with a 200mm lens. Shooting deer in Richmond Park is about as tricky as shooting cattle. I was sitting in plain sight on the other side of Beverley Brook. One stag kept looking at me. So whilst shooting the others I kept a little eye on it. Sure enough it came down. I thought it would drink but really, it wanted to cross. It joined me on my side of the bank. Fed a little but kept looking at me. It wasn't alarmed as it didn't snort. But it was uncomfortable with me close, so I withdrew as it's his park, not mine.
They don't behave like proper wild deer. Keep the camera in your lap and they watch you. Point the lens at them and they put their head down and continue to feed. Wild deer would flee as you brought your lens up, no matter how far away you were.