Is this sentence grammatically correct and natural?

"If I were a bird that *has* four wings, it would be fantastic."

See a translation

“If I were a bird that had four wings, it would be fantastic.”
You already used ‘were’ which is past tense so you have to use ‘had’ here. However ‘would’ needs to stay future tense here, like you originally have it. This is a confusing sentence because what you’re talking about isn’t necessarily in the past but when talking about if you had something or if you were something is always past tense (for some reason).
Examples:
“If I were you I wouldn’t do that.”

“If I had a million dollars I would spend it on traveling around the world.”

I hope this helped, the tenses in this sentence are quite hard!