Trenitalia pass and Bernina express

As far as I can tell, the Trenitalia Pass is only valid on Freccia, FrecciaLink, InterCity, InterCity Night, and EuroCity trains within Italy. The train from Milan to Tirano is a regional train operated by Trenord, so does not qualify for the Trenitalia Pass. It certainly does not apply to the Bernina Express.

short answer - no, sorry

You can save money by using the Swiss regional trains that travel on the same tracks as the Bernina Express. "Express" in this case does not mean fast; it is a tourist train. Using the regional trains actually took less time that the Bernina Express. The regional trains tend to be uncrowded, as opposed to the often filled Express, allowing a passenger to move from one side of the car to another as the scenery changes. If you are doing the entire route, you will need to change regional trains but they are scheduled so it's an easy transition.