"who claims the dependent that does not live with them."

"Claims" in 2020 is what matters.

"I assume the custodial parent received the additional stimulus also."

Not claiming in 2020 is what matters. Custody is not the same as claiming the dependent.

Perhaps it would help to review what is really happening:

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.

You might want to bookmark these links and read the IRS guidance.

Interactive wizards portal for determining dependency:

https://www.irs.gov/help/ita

And:

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.