I think the likelihood of the birth certificate being kept secret long enough for Meghan to pop out another baby and use the same hospital is vanishingly small, so like Iluvbertie, I don't see the point of this coyness about the birth certificate. It seem like a very odd and irrelevant thing to get hung up about when it's all public anyway.
Then perhaps it is to keep them from digging into the medical team right now at the peak of the story. They actually found the doctor that delivered Meghan 37 years ago to ask questions. Imagine that. Of course nothing was shared because of HIPPA. But they seem desperate. 42 days is a long time in today’s world. They might find something else to obsess over then.