I wanted to provide an update on this in hopes it may help someone else with the same issue.
Azure support team worked with me and we tried several things which continued to produce the same error. I was then asked to create another recovery services vault and use vault credentials file from the new vault to register the server. This was the solution which fixed the registration issue. Backups have been running for a few days now.
I still do not understand what occurred with this server that prevented it from registering with my main recovery services vault however.
Thank you to @SadiqhAhmed-MSFT for helping me to get in contact with Azure support team!