Doesn't say anything in the Constitution about that registration process. That's just an administrative procedure. Could that really overrule a Constitutional right? The lawyers could have great fun anyway.

What's the case law on this? Surely someone must have tried it by now, and appealed against any decision based on that administrative process.

In Irish law, for comparison, they might like to have children born abroad be duly registered, but if that hasn't happened it doesn't affect the right to citizenship based on an Irish parent or grandparent. And if they tried to bring in such a requirement it would be thrown out as unconstitutional.