Someone already worked out how to do it: IP over Avian Carriers. The ping time is terrible though.
Where do you see low bandwidth? Maybe over long distances but if I can give a pigeon a 1TiB USB stick and send it to the next city I bet it will be faster than uploading the data. If it arrives that is
People seem to always forget, that pigeons only know their way home. So you still need to get your pigeon(s) to the person sending the data with the pigeon back to you, in which case you can just pickup the data now that you’re there yourself.