This isn't a full technical answer, more a series of "I failed too" anecdotes, but 'answers in comments' are frowned upon [comment now deleted], so here we go…
I feel this should be read sitting down, with a large scotch. Sorry.
Even without the lighting issues, small animals/birds from a long way off are going to make disappointing photographs.
Either get close enough… under 10 feet, preferably under 3 -
or try to make something out of the overall setting & forget the detail of 'a bird' -
I'm not sure either of those are really keepers, but from last week's excursion, so fresh in my memory. These are both on a 300mmm, APS-C. Magpie & cormorants, if you've not got your glasses on ;)
Neither have had any significant work in post, as yet.
As you have a range of lenses to try, I'd at least go for the longest, the 500, at least once. You're not going to get anything, backlit or not, at that distance on a 200, whatever you do. I'd also go for extended exposure time over pushed ISO any day of the week on any bird that does 'staying still' for a living.
A tripod is going to help a lot. Lower ISO & longer exposure times will reduce the noise. There are arguments against that, but experience tells me the top half of the ISO setting is 'marketing ISO' rather than useable ISO. If you are relying on auto exposure to get a backlit shot, then override it with a good amount of compensation, or you will always get a black blob on a mid-blue sky.
Overall, I'd say that's a shot you're never going to get. The amount of ISO push or exposure push you're going to have to do to even get the colour of the owl into recognisable territory will be far outweighed by the disappointment at what it looks like when you crop in far enough to tell what it's a picture of. Try zooming on my cormorants for a good example. Sure, you can tell they're cormorants, but cropping to one individual bird is a pretty disappointing experience [even on the much larger original].
Flash has been mentioned, but if you've ever tried flash on a cat at night, then an owl will be a magnitude worse. Their eyes are light-catchers, they can see in the dark. Throw a flash in their direction & it will look like a bad horror movie.
That is, of course, if it doesn't simply fly off at the first flash.
…and note… even the best model can blink at just the wrong moment ;)