Wow, sounds like a great trip! You're already seeing Eagle's Nest, so I hope the tour has enough time planned. We spent two hours up there and I wished we had spent another one or two. Had a great time and a nice meal. Also had a nice meal on the lake (Konigsee) at the bottom south of Berchtesgaden. Wish we would've had more time in Salzburg, but you're doing it right.
A good day trip from Innsbruck may be Schloss Neuschwanstein and Schloss Hohenschwangau. Great castles to visit, and it's very close to Fussen, Germany, where many scenes from "The Great Escape" were filmed. Also, if you like heights, Zugspitze is a nice visit as well. (You're already taking a day trip to Italy, but another to Venice would be amazing. We spent a few nights there in October and loved it.)
We also wished for more time in Vienna, which was also very scenic. Only did the day tour at Schonbrunn, and wished we had more time to spend on the grounds. A meal high above the castle grounds in the Gloriette was memorable.
Seeing Bratislava from Vienna is very doable, crossing off another capital city. We spent a brief day in Vienna traveling from Auschwitz to Salzburg. A brief AirBnB stay in the city was all we had, so we chose to see Schonbrunn on our way to Salzburg. (My son streamed the PSU-NW game while we were driving to Salzburg.) Auschwitz is about four-five hours drive from Vienna.
The train system makes a lot of options possible. A quick visit to Prague or Budapest via the rails is doable as well.
Plenty to see in Munich as well. Schloss Nymphenburg is a good tour, and don't miss the glockenspiel show in Marienplatz. We had a great meal in the Ratskeller in the square. The camp at Dachau is about 20-30 minutes by their train system.
Hope you have a great time!