@GIS_SolutionsI've moved this post out of Ideas into Questions. The error message you are receiving is consistent with registering a feature dataset containing a UN without using a branch versioned database connection as @JayantaPoddar noted.
When registering the dataset containing the UN as versioned At any point did you see this dialog, and if so, do you select the check box and click OK (Note: this option is not supported for datasets like the UN)?
Can you confirm the Geodatabase Connection Properties for this particular database connection are now set to Branch as shown below?
On the other dataset that you are able to register as versioned, can you confirm it is registered as branch? (Right-click feature class and open Properties, view "Versioning". Does this read "Branch"?
If yes to all of the above, does the issue persist after closing and restarting ArcGIS Pro?
It is possible you are running into a workflow related issue caused by an initial caching of the register as versioned using move-to-base. In testing I ran into a similar issues but was able to get past this by closing and re-starting Pro. I am going to look into this issue further.
Jon