Dual-citizen baby name - how exactly should the names match?

Hi, my baby is a dual citizen of Korea and an EU country. I came up with my baby's Korean name after her birth and registration, so her EU country ID/passport doesn't include the Korean name.

It's a common practice for overseas Koreans to give their children a Korean name as a middle name (for example, Grace Jiyoung Kim). But I realized that it'll be a bureaucratic nightmare to change her legal name across two EU countries we split our lives between. My partner is currently involved in a massive legal procedure, and doesn't want to take care of this.

But I still want to give her the Korean name, at least for her Korean identification documents. According to my search, the most realistic solution will look like the following:

EU country ID/Passport: Valentina Kim Rossi

Korea (출생신고서 Birth Certificate/주민등록증 State ID): Kim Jina

Korea Passport: Valentina Jina Kim Rossi (the Korean government is quite flexible in terms of long Roman Alphabet names like this, but it still requires exact Korean name from the Korean State ID to be included within her passport name.)

So the question is, is it okay for her names to be slightly different like this? Looks pretty minor to me, but I heard that it might cause problems when you need documentation from country A in country B.

I’d appreciate any experiences or advice, thank you in advance.