@Mr. Peterson
In a hybrid organization, if you send email from Exchange on-premises, email will be sent from Exchange on-premises to the Internet directly.
So, I would suggest you create a new send connector for Exchange on-premises: Create a Send connector in Exchange Server to send mail to the internet
Then enter this external email address manually (Don't use auto-complete or reply old emails)
If you still cannot send emails successfully, I would suggest you try to use Message Tracking Log to check which step has issue:
Get-TransportService | Get-MessageTrackingLog -Sender user@domain.com -Recipients externaluser@domain.com -MessageSubject "subject" -Start 09/22/2022 -End 09/24/2022
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