It is possible that the amount they are asking is an import tariff or some other type of government fee for bringing live animals into the country. You say the airport is trying to charge you, but is it Customs? I don't know enough from your post to give you more of an answer.
Customs will have to be satisfied.
I agree with my colleague. More details are necessary. And you have to act quickly because no one at the airport is going to be taking time to care for those birds.
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No one online can know what is going on in your case because online we cannot find out any details. You need a lawyer. Check with a lawyer in your locale to discuss more of the details.
Good luck to you.
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I agree with my colleagues - there is simply not enough info to answer your question, but you should get yourself down to the airport and do whatever you have to do to get the birds out of there.