I would change the code in one main way, and that is to test if the ContentControl ("CC") is locked. If it is, and you don't unlock it, then you won't be able to delete it.

It is also important to note that this will not work for a nested CC unless you unlock them too. You would need to add a test if there is one or more parents. The best way to do that is to call a Function that tests the a parent's ID. If there is no parent then it will error out. If there is, then unlock the Parent and pass the Parent CC (via the ID you just got) back to the function and keep going until you get to the top Parent Level (i.e. Parent_ID will be "" when the test for it fails).

Sub Copy_CC_Text

On Error Goto lblError

Dim bLocked As Boolean

Dim oCC As ContentControl

Dim oRng As Range

For Each oCC In ActiveDocument.ContentControls

If Selection.InRange(oCC.Range) Then

Set oRng = oCC.Range
If oCC.LockContentControl = True Then

bLocked = True
oCC.LockContentControl = False

End If
oCC.Delete
oRng.FormattedText.Copy
ActiveDocument.Undo 1
If bLocked Then

oCC.LockContentControl = True

End If
Exit For

End If

Next oCC

lblExit:

Exit Sub

lblError:

Debug.Print "In Copy_CC_Text got error: " & Err.Number & ", " & Err.Description
Goto lblExit

Exit Sub

BTW, I found that when you undelete a Date (calendar) control it doesn't come back as a date CC. It also doesn't remember the date format or storage formate. So I had to create variables that save my CC's type, date format (e.g. MM/dd/YYYY) and storage formate (for this later one, I often set it to just Date vs Date and Time if it is just a date)

So at the top in my Dim section I added:

Dim sDate_Format As String 'In case we have a Date control we will have to reset its format after it is undeleted
Dim sDate_Storage As String
Dim sType As String 'Need this because we have to reset the type to Date after it is undeleted if it was a date CC

And then when I found it I added:
sType = CC.Type
If sType = wdContentControlDate Then
sDate_Format = CC.DateDisplayFormat
sDate_Storage = CC.DateStorageFormat
End If

Finally after I did the undelete I added:
If sType = wdContentControlDate Then 'If you check sType it will be 6 which is the Date type.
ActiveDocument.ContentControls(sCC_ID).Type = wdContentControlDate
ActiveDocument.ContentControls(sCC_ID).DateDisplay Format = sDate_Format 'We also have to rest date display, e.g. MM/dd/YYY
ActiveDocument.ContentControls(sCC_ID).DateStorage Format = sDate_Storage 'We also have to rest date storage format, e.g. date only
End If

I also wanted to note that I needed to put a DoEvents after the Delete as I found that date controls were not always being undeleted (and as I mentioned, it wasn't being undeleted as a Date control). It may be that was because it wasn't being undeleted as a Date control it had to take extra time to decide what to restore it as, and because on top of that my CC was nested 3 deep, or just because. In any event. Adding the DoEvents there seems to have solved that.

I also found that it was having problems with CC.Delete so when I found my CC I saved its ID to sCC_ID and then replaced CC.Delete with:
ActiveDocument.ContentControls(sCC_ID).Delete