There is no legal way to do it besides an amendment that annuls it.
However, that didn’t stop trump from trying to declared Obama illegal by stating he wasn’t born in the country. It totally ignores the fact that anyone born to an american citizen becomes a citizen. But who cares about facts.
I can see quasi-legal and illegal methods being used and accepted because the supreme court says they are legal. After all the court can make judgements that are totally wrong, like the dred scott judgement. Imagine that a law is passed that states no illegal can get medical attention from hospitals anymore. Hospitals are the usual providers of birth certificates. W/o such a certificate, one could be declared illegal.
And I have heard many republicans demand that health care be denied to illegals if they can’t pay for it themselves. It is not inconceivable that states or the federal government could pass such laws And republican courts support them.
Another is to declare illegals as criminals who have no rights and that is passed down to the child. It has been done in the past in the US, just not for a long time.
It could take years, if not decades for someone to prove they were a citizen if methods were employed to deny that to them. None of them might be legal methods, but if the courts went along with it, it could be protected. I look at hobby lobby decision, the baker’s decision, citizen united, etc such things could be protected by the court.