I will give you the advantages to starting in Venice, then you can evaluate them against the opportunity to experience the Angelus.
Venice is quiet, small, low-key sights, a great place to relax and soak up the atmosphere while you get over jetlag. Getting lost in the back canals is a joy. Rome is the complete opposite: a big, noisy, bustling city with major sights and getting lost, while not quite as easy as in Venice, is very frustrating and usually means wasted time and more walking than you'll want in the heat of August. Getting to the airport in Rome is easy and not expensive, even for an early morning flight. Getting to Marco Polo will usually take a long time or can be pretty expensive - though if there are 3-4 of you, a water taxi won't be too bad.
If you want to see Pisa, it's quite easy to stop on the train between the Cinque Terre and Florence. There's a left luggage room at the train station. It's a short taxi ride to the Leaning Tower, then you can walk, take the bus or a taxi back to the train station. If you want to climb the tower, you should buy tickets in advance - they often sell out. Or you could day trip from Florence, even combine a 1/2 day there with a 1/2 day in Lucca.