The lenses that werds suggested seem like sound suggestions to me, considering your 800 pound budget.
Just keep in mind that even at 600mm, or 800mm, you still have to get pretty close the animals in order to get "frame filling" photos. A lot of people think that long telephoto lenses are far more powerful than they really are. No camera lens, not even a 1200mm, will enable you to shoot frame-filling images of distant wildlife.
Once I had a guy stop and ask me a question at a National Wildlife Refuge. I was shooting some deer that were about 250 yards away with my 800mm lens. He saw the big lens, and he saw the deer out there in the prairie, and he asked me, "So, what are you taking pictures of, deer eyeballs?"
I said, "excuse me, what do you mean?"
He replied, "With that huge lens you must be getting closeups of their eyeballs, right?"
I responded, "No, not at all. In fact, these pictures don't bring the deer very close at all."
He thought I was joking, or being deceitful; he just couldn't believe what I was saying. So I invited him to look thru the lens. He came over and looked thru the viewfinder, and was shocked. "I can't believe it! I thought with that big lens you'd be getting closeups of their eyeballs."
In reality, what he saw thru the viewfinder of that 800mm lens was 4 or 5 deer that fit into the frame easily, with room to spare on either side of the herd.
I have had many people say similar things to me throughout the years. It just shows me that many people who are not familiar with wildlife photography or long telephoto lenses have a complete misconception of what long lenses are capable of. To take nice wildlife portraits, you really need to be close to the animals, no matter how long your lens is. If an animal is truly "distant", which is the term you used, then you will not be able to get portrait style photos no matter what lens you are using.
"Your" and "you're" are different words with completely different meanings - please use the correct one.
"They're", "their", and "there" are different words with completely different meanings - please use the correct one.
"Fare" and "fair" are different words with completely different meanings - please use the correct one. The proper expression is "moot point", NOT "mute point".