Sleeping in Milan

I’ve stayed at Hotel Gran Duca di York which is quite near the Duomo. I don’t know if it meets your criteria for moderately priced. I did not look to see if they have vacancies for your dates. I know they are filled for my dates later in September.

https://www.ducadiyork.com/en/hotel-milan-city-centre/1-0.html

Hotel Berna near the central train station is often mentioned as a favorite of forum members.

The Westin Place is comfortable and air conditioned, which is critical in Italy in July. Plus, it's expensive enough to keep out the riff-raff.

We enjoyed our stay at Antica Locanda Leonardo very near “The Last Supper.” We walked to the Duomo but there are teams too.

Plus, it's expensive enough to keep out the riff-raff.

My wife and I prefer to stay in modest B&Bs. It has led to wonderful experiences, not the cold, impersonal experiences we've had in expensive hotels.

As a bonus, it keeps us away from the kind of people who try to avoid the riff-raff.

Hotel Berna near the central train station is often mentioned as a favorite of forum members.

Great breakfast and close to the railway line to Varenna-Esino, but what would they do 2 evenings near the central station?

I'd rather stay in a nice area and take the metro to Centrale on the departure day, better than wasting 2 nights out in a deserted business district.

Many thanks for everyone’s replies and suggestions. I am happy to report I have booked at Anitica Locanda Leonardo and am very pleased with the ambiance it projects!