The carrier isn't of note any more so than the car seat the kid goes home in is. When it comes to birth certificates, the new paradigm should link the certificate to a file, with all pertinent genetic contributors listed there, but for the sake of travel, passports, etc., the parents who will be raising the child should be on the birth cert.
Sounds a little too bureaucratic and impersonal, not to mention possibly hard to access the information.
Travel docs don't matter, the child should be the major concern as I see it. A pasport is different, it is not the birth certificate and another issue entirely. Let's stick to the topic shall we.