Shooting legal expert
Bar-headed geese are uncommon in the UK. They are natives of Central Asia and in winter migrate to India, sometimes flying at 27,000 feet over the Himalayas.
A few are kept in collections here from which they occasionally escape.
We had a couple on the Thames a few years ago but they have now disappeared.
The Wildlife and Countryside Act regulates the shooting of wild birds but birds raised in captivity are not covered by the act. Broadly speaking a wild bird is one that has a self-sustaining breeding population in the wild or is a bona fide migrant visitor.
The bar-headed goose is neither of these although a few have been recorded as breeding.
They come in Category E, the lowest category, of the British List.
Im not aware of any legal precedent in relation to the accidental shooting of bar-headed geese but I dont believe you have committed an offence.