There appears to be a bug with creating labels for web layers when they are first added to a Pro Project sometimes from AGOL when opening a web map. I have even had layers where a label class was enabled in AGOL and when the layer is loaded into Pro, the label class is not accessible.
Instead of storing it locally, you can copy/paste the same layer into the project then remove the problematic layer. This has allowed others and me to create label classes this way while still maintaining the connection to the hosted service.