But "He feels himself a failure" is in my opinion perfectly correct and idiomatic, although it is something I (72 yrs) would be more likely to say than someone half my age :) But I think it should be accepted, and I am a Native English (UK) Speaker (living in Finland, fluent in Finnish, learning Swedish for fun).
This could be rendered "He feels himself a failure" which is pretty idiomatic but a tad more formal than "He feels like a failure;" or could be "He feels himself to be a failure," which is correct but slightly stilted, a flaw shared by the terse but also correct "He feels a failure."
It's a perfectly fine phrase, of course, but it corresponds better to a different one in Swedish - for instance, Han ser på sig själv som ett misslyckande. Note that one sentence uses an adjective, and another a noun. We do not generally allow for translations that change too much into something else, and though I absolutely get your argument, this is a very good example of that. Obviously, in real life, your translation might have worked great.
Also, I'm not a native English speaker, but wouldn't you say that feeling like a failure is a little more temporary, while considering oneself a failure is more like a permanent state?