You could create a vector tile service in your coordinate system of choice to use as a basemap.
Insert a New Basemap
Choose data in the coordinate system you need and set the basemap coordinate system to match it.
then share this layer as a web layer in the vector tile format.
Then add this service to your basemap and remove the original source data.
This will give you a basemap to use in the coordinate system you want.
Add operational layer to your MapView and select the basemap that is using the Vector Tile layer you created. Set your maps coordinate system to match the basemap.
When you share this as a webmap, No errors about coordinate system or basemap.
And it looks like this on ArcGIS Online in the MapViewer
Mark