I’d try Spot AF. Use a lens with the widest aperture possible, and assuming there’s a fence between you and the subject get the front of the lens as close to it as possible.
The following were in enclosures, the first netting, the second a wire mesh. You can see the effects of the enclosure barrier, but they are mostly blurred out.
"Wattled Crane"
EOS 5D Mark II, EF 70-200mm f/2.8
L II IS USM + EF 2x II Extender @ 260mm, 1/250 s, f/5.6, ISO 400
"Bald Eagle"
EOS 7D, EF 70-200mm f/2.8
L II IS USM @ 200mm, 1/250 s, f/2.8, ISO 200