Looks like I can rent the Sony 200-600mm lens for $105 locally or on borrowlenses.com. I may do this.
Regarding zoom vs. fixed -- what mm lens would you get? I'm thinking the versatility of being able to go from 200mm all the way to 600mm means I'd be able to tightly frame my subject no matter the distance. The only fixed lens I ever use is an 85mm that I use for portraits. It's amazing, but I use it in much more controlled environments.
I'd opt for a 400mm f/4 and carry a teleconverter if needed, but this is a personal preference as well. And it's not like I'd be using all 45 megapixels so if it needs some post-production cropping, I'd be alright with that as well. I'll say that I shot a 100-400 f/5.6 for sports and zooming + focusing just took too long. f/5.6 and higher just wouldn't cut it unless there was full sun.
Just some personal preferences, I've really got no clue about what Sony offers but it seems consistent with other lens makers: full-frame glass is expensive.
Rent that 200-600, maybe it's all that and a bag of chips!