Submit both and explain the difference truthfully in your filing. Do not choose the more convenient document or adjust a spelling to match. Where a birth place appears under a former country name, include the record showing it and any official document connecting the two versions of your identity.
Explain discrepancies with authentic supporting records
Imagine a birth certificate recording a city under a name the state no longer uses, while your passport shows the current name. That is a documentary difference, and it is resolved with an explanation and supporting records, not by editing anything. If a government notice later states a birth place or classification you believe is wrong, investigate the conflict and respond through the official channel rather than repeating information you know to be inaccurate.
The program’s vetting stage will require identity and background information for each applicant. Vetting is where personal history matters: identity, background, prior immigration history and the lawful origin of the funds used for the gift are the kinds of facts an applicant should expect to document. In the meantime, do not send passports, financial statements or identity documents to an unverified recipient while the approach is being assessed.