It's a limitation of the web map that it can't adjust to changes in the REST endpoint.
However, I believe if you saves the popups on the item I think it should be fine. There is an option in the layer context menu to "Save Item Properties" In general, this is a better workflow because if you then add the item to another map, you don't need to define the popups (or symbology) again as it is saved in the item. If you find any issues, post it here.
thanks,
Mike