Currently it is not possible to directly overwrite an ArcGIS Server print geoprocessing service from ArcGIS Pro - this from discussion this week with my local ESRI technical support. When layout names change or layouts in a registered folder (which I have) are added the service needs to be republished so that web applications can see the up to date definition and make use of new/changed layouts. Print services should not change all that often, however they do still need to change, and at the moment my workaround is to delete the service and then publish again with the same name to avoid having to update URLs in web apps (namely LocalMaps). Overwriting the service in-place in the same way as overwriting a feature/map/etc. service would be great and save time as well as clicks.
I have just upgraded to Pro 2.5 and used it to successfully publish as per the workaround I mentioned.