Here are the steps I am doing to publish a web map which I want to be my base map using my spatial reference - NOT esri's
How do I change this setting so I can create a basemap as a web layer using my spatial reference?
You are correct with your last point you will want to publish a tile layer with your desired parameters to Arcgis Online.
Once you have published it to ArcGIS Online you can follow the below blog to create your own custom basemap gallery including using the imagery that you have just published as a basemap.
Additionally, if you are publishing large amount of data you should use the recommended workflow found below under the subsection "Publish large tile packages"
Thanks for the detailed reply and the helpful links. I should have mentioned that I'm using AG Enterprise, not AGOL. In reading your links, it seems to me there are so many options (and so little time).
I'm afraid there is no solution to change the Tiling Scheme when publishing a web map from ArcGIS Pro. The software must assume all data is in "ArcGIS Online / Bing Maps / Google Maps."