There isn't really a "Claim" for the Payments. What you have is Advanced Payments made in 2020 against best guesstimates from 2019 or 2018 tax returns, and Reconciled to Actual Eligibility in 2020 via the 2020 tax filing.

"that because the baby was born in september, after the cut out date of July 15th, she would only qualified for the second stimulus for the new baby. Is that new?"

That would apply to the timing of the Payments. The Payments are In Advance.

The stimulus formula is right here.

The funds were paid out as Advanced payment against a projection. The projection used 2018 or 2019 tax returns. But 2020 is the Actuals. You use the 2020 return to reconcile what a person is entitled to, against what they got. There is no splitting of 2020. There are two amounts, but no split.

https://www.irs.gov/newsroom/economic-impact-payment-information-center-topic-a-eip-eligibility

https://www.irs.gov/coronavirus/second-eip-faqs#Eligibility

One for each EIP.