Should I Change My Daughter's Name if It Sounds Like a Name With a Bad Meaning?

After reading your full situation, everything is clear. Your concern about your daughter’s name “Airah / آئرہ” is understandable, because in Islam a good and meaningful name is preferred. However:

“Airah” is not a proper Arabic or well-known Islamic word that has a clearly bad meaning. Words like “عائرہ” or “عاہرہ” are different. Just because they sound similar does not mean the same ruling applies. Scholars (especially in Hanafi fiqh) say a name is considered bad only when it clearly has an inappropriate or sinful meaning. Otherwise, it is permissible.The positive meanings found on Google (like “honorable”) can also be considered, and such names are treated like many modern names that don’t have a fixed meaning.

If the country’s law makes changing the name very difficult, then Islam takes such compulsion into account. There is no sin on you. In fact, your effort and concern show your good intention, and Inshaallah you will be rewarded.

Advice:

Keep “Airah / آئرہ” in official documents. At home or among family, you can call her “Azka” (which means pure and pious, a very beautiful name). This is completely allowed, just like a nickname. Many people do this. Slowly make your daughter familiar with both names.