Canon lenses hold their value, so I have stayed with their offerings except for a Zeiss 18mm super wide. I own both of the Canon 70-200 IS lenses, (2.8 version II and f/4), and each has their place. Outdoors in the daylight the f/4 is just fine, though the IS is noisy compared to the f/2.8. Both take the 1.4X TC version III quite well, and it doesn't weigh much or extend the lens enough to worry about. With your budget, go for the Canon f/4. You will be able to get your money out of it if you later want a faster lens or longer zoom.
The f/4 isn't as sharp wide open as the f/2.8, but it is half the weight and half the cost. The f/2.8 focuses faster than the f/4, if that is an issue for your subjects. I will take the f/2.8 on a 4 mile hike, but on a 10 mile trek it would be a bit much. The 2.8 is a phenomenal portrait lens, and I find the weight hand held a non-issue. I have never used it on a tripod, the IS is that good. For indoor events the f/2.8 is worth its weight in gold, pun intended.
While I started with crop bodies, (60D & 40D), once I bought a used 5D Classic I have used the crops much less. The focal lengths of 24-105 and 70-200s are just more useful to me with full frame. When shooting wildlife (raptor or antelope for example) I need more reach than even the 1.6X crop factor. The 400mm f/5.6 is my go to lens for these subjects. If you get a used 5D Classic, pick one with a serial number that starts with a 2 or 3 so you have the better LCD screen. After getting spoiled with FF, I bought a used 6D, and I couldn't be happier with its all around performance.