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This is a very difficult one, Kitty. Firstly, I presume, since she is a British raised child, that the East Indies referred to on her birth records is India, not Indonesia, as the British were only ever in Indonesia for a few years during the Napoleonic Wars (well before 1822).
Your only chance, and it is a slim one, is to search through the records of the British India company, in the hope that you might find a matching birth to a father named Alexander, but it is a long shot. Swarthy doesn't necessarily mean she had a non english father. It just means dark complexioned. If she were negro or indian, they would probably be more specific. I would tend to look under the name Atkins, and presume dad was alexander Atkins. You know there were brothers. If you can find four children born to a father named Alexander Atkins, you are home. Hope this is of some help. BTW SOme British India recs are online.