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Within Europe (I'm assuming the story is set in somewhat distant past) names would generally be translated, especially if a member of royalty assumed a new position in a new country. Even now prince Charles is Tywysog Siarl in Welsh. So it stands to reason that the hypothetical Irish princess's name would also have been translated by speakers of Welsh. Or at the very least "welshified" - made to be possible/easier to pronounce in Welsh (the same way Yanmei is already kind of translated because first it's not the original spelling and second it doesn't account for tones).

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I can't help with the actual problem though. The first thing that comes to my mind is finding an incredibly widespread name that exists in both Irish and Welsh (like Jane: Síne and Siân) but it's kind of cheating.