The Sigma 150-600 is excellent value for money, it really is in my opinion, the best 600mm lens for that kind of money. I have one, it's super sharp. The only crit I have is the aperture at 600mm, being f6.3, I often struggle with that but I'm usually under heavy tree cover or undergrowth. Give it the light though and it's fine.
I can't really advise on a body as I'm not Nikon. I use both full frame and APSC and I find myself reaching for the crop sensor (M5) more than the FF (5Div) mainly because, even with the Sigma on, it's quite easy to carry around. I like the extra reach too but the image quality isn't as good as the 5D. I get 960mm equivalent on the M5 with the Sigma. I needed the extra reach as I wasn't quite as close as I wanted to be to my subject (kingfisher) as I wanted to be to begin with but towards the tail end of last year, I was able to get close enough to use a full frame camera. Just a few feet make all the difference and I eventually was taking pictures of perched kingfisher from about 15 feet.
In your position, I'd go for a crop sensor. That said, images don't have to be frame fillers, they can show the habitat too.
There are a few ways of photographing kingfisher. Paid hides make it as easy as it's going to get. I have nothing against paid but you do miss out on the experience of prepping your own set up and the knowledge that brings. It takes time that way, first you have to find a kingfisher and then it's usually a case of watching it and working out its habits, which can take a while. You also, in most cases, unless you have a kingfisher that is used to people, cars etc, have to know how to hide yourself. That will come though. This would apply to wildlife in a lot of cases, not just kingfisher.
Patience is key too.
I wish you well, I really do, be prepared for some long days but when you nail it, you'll know why you do it.
