The 200mm f/2.8L II is a very nice lens and reasonably compact, nice build, great image quality.
I don't have one. That's because I have the 135mm f/2 and a good 1.4X.... which gives me a nearly identical 189mm f/2.8 combo.
For wildlife on full frame.... 200mm ain't nearly enough lens.
On sale right now for $500, the EF 70-300mm IS USM II is quite compact, has decent performance and IQ. No, it's not an L and doesn't have f/2.8... but when you get into fast, long lenses you need much deeper pockets. The 70-300mm on FF is kinda short for wildlife, but a lot better than 200mm. Also, this lens really can't be used with a teleconverter. The Canon TCs won't even fit. A Kenko 1.4X would fit, but you end up with an effective f/8 lens that your camera won't be able to autofocus. The camera's viewfinder will dim down, too.... making manual focusing more difficult (Live View with Exposure Simulation can help brighten things up, but is pretty slow to shoot with).
This is one of the advantages of the Canon EF 300mm f/4L IS USM.... it can be used with a quality 1.4X teleconverter and the effective f/5.6 will still autofocus on all Canon cameras (though I think your camera might be limited to the center AF point only).
The Canon EF 100-400mm IS USM "II" is way more than what you want to spend at just over $2100. But I've seen some used copies of the first version (push/pull zoom) of 100-400mm selling for around $800-$900. They are still available new, too... but are about $1250.
Canon also makes an EF 400mm f/5.6. USM that's reasonable size, not too expensive (might find one used), has fast autofocus and is quite sharp. One problem, it lacks IS and stabilization is a very nice thing to have on long telephotos like these. I prefer the 300mm f/4L and 1.4X TC, which gives me an effective 420mm f/5.6 with IS. The shorter lens with TC also gives me choice of two useful focal lengths, which can be handy, although not as fast,convenient and versatile as a zoom. I've bought two 300mm f/4L IS USM used for around $800-$900, if memory serves. The Canon 1.4X II can be found used for around $250. Alternatively, the Kenko Pro 300 1.4X DGX can be found for around $125 I think. I've heard it's quite good, though I haven't tested it myself. (And I'd rather use a 1.4X with a 300mm than a 2X with a 200mm... 2X tend to degrade IQ a lot more than 1.4X.)
Sigma just recently introduced their own Sigma 100-400mm OS HSM, which is selling for just under $800 and is a little smaller than the Canon, but only a few ounces lighter weight. One problem with the Sigma that I see is it's lack of a tripod mounting ring or any means of optionally attaching one. It's okay if a smaller, lighter lens like the Canon EF 70-300mm IS USM II doesn't have a tripod ring.... but not a heavier, larger lens like the 100-400mm! I think Sigma made a mistake, not at least providing means of adding an optional tripod ring. The Sigma 100-400 on a camera with a tripod or monopod attached via the camera's socket will be very unbalanced. I[m sure it would cause many tripod heads to "droop".
But Tamron has also just announced a 100-400mm VC USD lens. There's not a lot of info about it yet, but it's expected to be priced to compete with the Sigma. One nice thing about the Tamron is that it can optionally be fitted with a tripod ring. Tamron plans to sell one separately and it even has a built-in Arca-compatible quick release dovetail. I have no info about the cost of the optional tripod ring. But IMO it makes all the difference with a lens the size and weight that the 100-400s need to be. These lenses are hand-holdable for a reasonable length of time... but much more than an hour or two, you're likely to want a tripod or at least a monopod.
Yes, there are also the 150-600mm Sigma and Tamron lenses. With your limited budget, you are looking at the Sigma "Consumer" version (not the "Sport", which is about 2X the price).... or the older original version of the Tamron. Both those lenses are pretty good up to 500mm... but beyond that they fall short of the newer and more expensive versions of 150-600mm and some other 500mm and 600mm lenses that cost a lot more. These also are pretty large and heavy lenses (about 50% heavier than the 100-400s). With them, you will probably want to use a tripod most of the time.
Several people have mentioned and shared photos made with APS-C cameras. Those have an advantage for long lens work... allowing you to use smaller, lighter less expensive lenses. I use a pair of 7D Mark II for a lot of sports and wildlife, often with 300mm f/4L IS USM or 100-400mm II lenses. The APS-C cameras give a "free 1.6X teleconverter" effect. By "free", I mean there's no light lost, the way there is with an actual teleconverter. Using the same 300mm lens on our cameras, you'll get a 20MP shot with 300mm, while I'll get a 20MP shot with a lens that's acting like 480mm would, on your camera. Or, in other words... If I wanted to frame a subject on my full frame 5D Mark II the way as with a 300mm f/4 on my 7DII, I'd need to get out a 15 inch long (plus hood), 8 lb., $9000 500mm f/4 lens.... and a tripod to sit it on!
I'm not saying it "can't be done".... In fact, these images were both shot with 300mm lens on my 5DI:
The young black-tail buck was shot with 300mm f/4L IS USM plus Canon 1.4X II teleconverter and the first image isn't cropped significantly (just to 5:4 aspect ratio). The second image is a crop detail from the same shot, to give you some idea of the IQ (as best we can at Internet resolutions. The third image is another shot taken close to the same time and not cropped other than to 5:4 aspect ratio... yeah, I gotta kinda too close!
The red tail hawk was shot with just the 300mm f/4L on 5DII and I needed to crop that image quite a bit more. The two smaller images are details cropped from the large one. I was very lucky that hawk was busy hunting, flew by a number of times and gave me about half an hour of photo opps... 5D Mark II is not the greatest camera to shoot fast moving wildlife. I probably took 100 or 200 shots to get a few "keepers".
While it's possible to get some wildlife shots with 300mm on full frame, you'll have a lot less opportunities than if you had a longer lens to work with.