Unsure what the best course for action is at the moment. But one idea I did have is use a OnErrorResumeNext class flag:

class Class1
  Private LastErrorText as String
  Public OnErrorResumeNext as Boolean
  Public Property Get ErrorText() as String
    ErrorText = LastErrorText
  End Property
  Sub Error1()
    Call RaiseError("Error1")
  End Sub
  Sub Error2()
    Call RaiseError("Error2")
  End Sub
  Private Sub RaiseError(ByVal message as string)
    if OnErrorResumeNext then
      LastErrorText = message
    else
      Err.Raise vbObjectError, "Class1", message
    end if
  End Sub
end class

Sub ErrorInClass()
  Dim c As New Class1

  'Raise a handled exception
  On Error Resume Next
    c.OnErrorResumeNext = true
      Call c.error1
    c.OnErrorResumeNext = false
  On Error GoTo 0
  'if needed check ErrorText
  Debug.Print "ERROR: " & c.ErrorText
  
  'Raise an unhandled exception
  Call c.error2
End Sub

Or, export this functionality to a seperate class:

Class ErrorHandler
  predeclared_id = true
  ...
End Class
Class Class1
  Sub Error1()
    Call ErrorHandler.last.Raise("Error1")
  End Sub
  Sub Error2()
    Call ErrorHandler.last.Raise("Error2")
  End Sub
End Class
Sub ErrorInClass()
  Dim c As New Class1

  'Raise a handled exception
  On Error Resume Next
    With ErrorHandler.Create()
      Call c.error1
    End With
  On Error GoTo 0
  'if needed check ErrorText
  Debug.Print "ERROR: " & ErrorHandler.last.text
  
  'Raise an unhandled exception
  Call c.error2
End Sub

At the very least this method allows usage of “Break in Class Module” and does keep the callstack mostly intact, while also allowing for resuming handled errors in classes…

2024-09-19 Edit:

Yet another alternative… If methods in the class have error handling themselves, these won’t cause code to break and errors to occur. I think I subconciously knew this but forgot to include in this post. So the other alternative would be, in order to make some errors optional and others not to return a result type or error parameter. I.E.

Public Property Get Name(Optional byref errorOccurred as boolean) as string
  On Error GoTo Catch
    Name = This.iaccessible.accName
    Exit Property
  Catch:
    errorOccurred = true
    Name = ""
end property

In this way, calling code can break if wanted, or continue if wanted:

Sub DisplayInfo()
  This.nameBox.value = acc.name 'No change
End Sub
Sub AutomateSomething()
  Dim hadError as Boolean
  Dim el as acc: set el = getElement()
  Dim name as string: name = el.name(hadError)
  if hadError then Call Err.Raise(Err.Number,Err.Source,Err.Description)
End Sub