I successfully identified the problem. It appears that some configuration properties were modified during the service upgrade from the S1 to S2 Tier.
Solution: I accessed the server via SSMS, right-clicked on 'Properties', and under the 'General' tab, I selected 'Show Advanced (All) Properties'. I then altered the properties - specifically, 'ExternalCommandTimeout' and 'ExternalConnectionTimeout' - changing their values from 36,000,000 seconds to 3,600 seconds. It seems that during the upgrade from S1 to S2 Tier, these properties were inadvertently set to invalid values, which were excessively large and exceeded Int32.MaxValue.
I was able to find references to these properties under tab 'Diagnose and solve problem' in Azure Analysis Services portal and the help options there!