I have lots and lots of thoughts on this. I've obsessively thought about this everyday for.... 8 years or so. I've wrote and this and deleted it multiple times now.
I'm of the mind to say you should be able to use it, with and asterix*
I have native heritage, like grandfather was tribesman. As in my grandfather was literally forced by US law to speak English growing up. Thus causing a break between his father who didn't speak English, and this was very literally an attempt to ethnically kill off another tribe. When I see Americans put things like the confederate flag in a dream catcher, that's what cultural appropriation feels like. Or naming military things after native names after all the US military has done to them. Yeah, fuck all that noise. The US continues to do these awful things to natives.
I know from having lived in Japan, that if you wear a kimono, etc, and you do so respectfully, they will be delighted, overjoyed honestly. Many expats take a legal alias of a Japanese name, or Japanified name too, some as soon as they get off the boat. Most cultures love to share their cultures, not all, but the vast majority. Part of that is name sharing too.
The problem in the US, because people (read white and black) can't seem to tell the difference between, Southeast Asian, East Asian, Pacific Islander, and Native American, and also lighter skinned Latinx we all get lumped together. Many of these peoples have been giving shit for their ethnic names in America. Many have changed them to face less abuse over their ancestry. I was even told my name was too white, despite I'm mixed with Irish. Just another excuse I supposed to racially other me and call me racial slurs.
I think you should be free to take whatever name you want. You're not free of consequences and you can never be free of those in this universe. If you choose to do this, then I think you should read up on and learn about how certain immigrants and long term residents have been given hell for their ethnic names in America. Japanese pronunciation isn't terrible hard - but you shouldn't slack on it either - that said, Japanese names stand out in America. However, tonal names in America are butchered, then mocked, and then often forcibly given more westernized names/nick-names. To the point, in some cultures, you'd pick a more western name before coming here because you know what hell awaits you if you use your given non-western name. You're going to be hit with questions for the rest of your life, and I think you should be educated and have the spine to tell the truths for the peoples who were given these names at birth. That said, I don't think Kai is only from Japan.
I choose my name with the full insight I was going to emigrate back to Japan, and am finally in the process. I'll change it over just before I go, and even I will still need to take an alias since Chinese character names are not legally allowed in the US, or any other scripts. (Tho they should be allowed if the US really thinks it is multicultural rather then culturally imperialist.) Even though the name I chose in the long run is Sumire, I still continue to use Violet here in the US, and given how only less than 6 people will still know me here when I leave, I don't really care if they use Violet instead of Sumire.