The firstname.lastname@example.org,works, email@example.com does not.
Do you mean if external senders send to firstname.lastname@example.org, they would get the NDR (error or bounce) message 550 5.7.54 SMTP; Unable to relay recipient in non-accepted domain.
While if sending to email@example.com, the email can be received correctly.
If this is the case, what is the accepted domain setting on your Exchange server?
Please make sure you have the domain in the error message added as an accepted domain (Authoritative domain).
If the above suggestion doesn't help with this issue, besides the accepted domains, please also post the following setting of the affected mailbox:
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