Follow me on this and tell me if this makes sense. My wife and I are planning to spend 10 days in Europe starting in Berlin on July 4. July 4-9 in Berlin. My wife wants to go to Florence too. Nice cheap evening flight via Ryan Air to Pisa on a Monday night July 9 and we can bus or train to Florence on Tuesday 7/10.....so one night in Pisa off to Florence until our return to Berlin on Friday morning from Pisa.
Is this too ambitious? Flights from Berlin to Florence are pricey and has a layover somewhere.
Go for it. Depending on what you like you may or may not just wish you had more time.
Timing sounds fine as long as you have plenty of time (an overnight) before your next flight departing Berlin. Or do you really need to return to Berlin? If you haven't yet bought the inter-continental flight, then look for a "multi-city" flight plan, into Berlin and home from almost any Italian airport.
I wouldn't want to fly from Italy to Berlin on a separate ticket on the same day as my transatlantic return flight. If something goes wrong with the Pisa-Berlin flight, you are going to get no help from your transatlantic airline and will probably have to buy a costly last-minute one-way ticket to get home.
Do try to fly into Berlin and out of a northern Italian city so you don't have to make a return flight to Germany. From my origin, fares to Milan are sometimes reasonable.