Germany, Alemania, Deutschland.
Mexico, Messico, Méjico.
Japan, Nippon, 日本.
China, Zhōngguó, 中国.
Every country gets by getting called different things in different languages. But Turkey insisting on the spelling just because they don't want to be associated to türkiye birds, where the birds are literally named after the country, is a silly move designed to stir nationalistic pride for their near totalitarian government.
I'm from a language where you'll never get it right, and most people are pragmatic and don't care about names being the way our mothers do. That's a very Western thing.
Forcing us to try to teach you our language, or half-way butcherings, are considered obnoxious. Go with the English pronunciation. Do anything half-reasonable-sounding, and it's okay too. Don't visibly struggle each time you try to say my name.
For my name specifically, there's also an uncanny valley phenomenon. My name pronounced obviously wrong by an American doesn't raise flags. Americans mispronouncing my native name sounds horribly wrong.
Similarly my given name has an English pronunciation that I prefer people use when speaking English. Someone trying to pronounce it natively without being native would feel extremely awkward and not respectful at all.
"Note: The official conventional long-form and short-form names remain “Republic of Turkey” and “Turkey”, respectively. “Republic of Türkiye” should be used in formal and diplomatic contexts. The conventional names may be used in place of or alongside “Türkiye” in appropriate instances, including U.S. government cartographic products, as it is more widely understood by the American public."
NATO head Jens Stoltenberg started saying Turkey but then changed to Türkiye before the big NATO summit in Madrid last summer.
As someone who was in that country for a year all over the place with a rifle attached to my chest, Americans always ask me about my exciting vacation there. As an American convervative myself, people's heads are blown whether the are left or right leaning when I pronounce it as "Ih-rahk." At this point I find the reactions fairly comical and get a kick out of experimenting on people, much to the dismay of my wife, who thinks I should just go with "Eye-rack" to stop embarrassing her.
By the way, I pronounce Turkey as like a Thanksgiving "Turkey."
Anecdotally I was told once upon a time by a first generation immigrant from Oman that you should pronounce a countries name as they pronounce it, as a respect thing. I thought more about it, and if I heard someone say United States as "Ooonited States," I wouldn't care one bit, but there really is only one way to pronounce the United States it so it's not really a fair comparison.