Let me know if remote debugging options is good enough for you.
You can also have a local backup of your Azure SQL database by exporting the database as a bacpac to a local drive in your computer. You can use SqlPackage utility for that:
SqlPackage /Action:Export /SourceServerName:SampleSQLServer.sample.net,1433
/SourceDatabaseName:SampleDatabase /TargetFile:"F:\Temp\SampleDatabase.bacpac"
You can use PowerShell also.
$exportRequest = New-AzSqlDatabaseExport -ResourceGroupName $ResourceGroupName -ServerName $ServerName `
-DatabaseName $DatabaseName -StorageKeytype $StorageKeytype -StorageKey $StorageKey -StorageUri $BacpacUri `
-AdministratorLogin $creds.UserName -AdministratorLoginPassword $creds.Password
After that import that bacpac to a local SQL Server instance.