Prepositions are tricky, as they do not always directly translate, as English speakers would use a different preposition to a German speaker. In this case, per Vogel might be the correct way to say it in German, but we would say by bird. I don't think per bird should be accepted, as it is incorrect English, but users need to be aware of what contexts German speakers use certain prepositions in that differ from us.
So does this connote that the greeting was "on" the bird, as in tied to it? Or does per mean here the more general "via" (which doesn't specify HOW the bird was carrying the message, but merely that it did).
No, I think 'per' has the same meaning as 'via' or 'by', it doesn't specify how it was carried, could be tied to the leg, held in the beak, or the bird memorised the greeting and told the recipients.
the alternative solution given is not correct English: you can't say "per bird" in English
what are they trying to say - per bird? I don't understand the meaning of the sentence...