@Michael Ryan Apologies for the delay, researched on this ask, I am not sure what will be the different between MySites from SharePoint if MySites part of SharePoint, would suggest to test this scenario and if it fails you need to work with SharePoint team ( via support ticket) to troubleshoot this further.
In my view this approach which you mentioned in your query should help same deployment procedure + custom domain registration of the subdomain to facilitate the site.subdomain configuration in Azure App Proxy.
Let me know if you have any further questions.