I just noticed myself doing something in my new story that I'm not sure I should be doing. When I have a character whose name that's pronounced differently than it's spelled, I find myself pointing out the right pronunciation in the story itself. Is that okay? Like... "My name is Knight Captain Cio," the armored woman said, pronouncing it chee-oh." (the Latin word for movement) or "You can call me Gyro!" "What, like the Greek food?" "No, like gyroscope. Y'know, cause I'm always flipping around and stuff? Yeah?" "Sure, whatever."
The question you need to ask yourself in these situations, is whether or not it matters how your reader is pronouncing it. If it doesn't then leave them to it. Or use names the pronunciation of which are in no doubt.
Cio might not matter, since not many people know Latin. But I think Gyro is important since it correlates with his powers, and I don't want them pronouncing it like the Greek food.
Well for what it's worth I didn't know it was a Greek food and was pronouncing it Gyro, as in the old benefit cheques (Giro) or the scope you mention.
100% with @OurJud. This particular flavor of narrative intrusion (that's what it is) comes up in questions with regularity. In the example scene you gave, the characters are no longer actually talking to one another. They're talking to the reader, imparting your thoughts, not their own. This came up just a couple of weeks ago... https://www.writingforums.org/threads/things-you-wish-authors-would-stop-doing.164684/page-4#post-1833952 https://www.writingforums.org/threads/things-you-wish-authors-would-stop-doing.164684/page-4#post-1833992
I had a story like this once with a bunch of frat bros. I didn't like how their frat house looked in text, so I used: ". . . that's Kappa Chi, rhymes with kiss the sky." They were bragging on their name while subtly sounding it out for the reader, and that worked really well. (I have a whole series of apocalyptic frat stories, and their tone allows this.) Anyway, it's kind of meta to mix that in early just to square aware pronunciation, but it works better than the phonetic spelling. I favor your second method.