External domain is not added/authoritative to your exchange.
So you need to add that domain as accepted domain and set it to authoritative false and relay should be external relay.
Then it should work so when you email to external domain user exchange understands that this domain is external to it and will send email from it to external domain.
Set-ReceiveConnector "<SERVER>\<ConnectorName>" -PermissionGroups AnonymousUsers
Get-ReceiveConnector "<SERVER>\<ConnectorName>" | Add-ADPermission -User 'NT AUTHORITY\Anonymous Logon' -ExtendedRights MS-Exch-SMTP-Accept-Any-Recipient
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