For the benefit of any others trying to solve this, I finally managed to get this working as part of the DevOps Release pipeline, but encountered many problems along the way.
Most notable is that if I tried to execute the Migrator exe as part of the Release agent tasks, it failed to access the Azure SQL Database - this is because it is running in the context of the Agent server, whose IP address is taken from a pool, and that pool is cycled weekly, so no chance of Whitelisting it in the Azure SQL Server firewall.
I got around this by creating a new MVC Controller in the Web.Host project, and added an Action that launches the Migrator exe. Then I created a PowerShell task in the Release pipeline that make a simple Http Get call to the new Action.
I'm also wanting to do exactly this.
Should I publish the Migrator app to the same Azure app server as the Middleware/API web app?
Then create and configure a Release task to execute the Migrator?
Many Thanks