I realize you might never have seen the Vatican and understand your desire to see it. I would not want to miss it either. This is a jubilee year, which is going to mean even more crowds than this past year. We were there this past Sept. and was so crowded that it limited my experience. I honestly think it might have been better on our own, not trying to keep up with the tour. Our guide explained why this helped with decision to not include the Vatican on some of the tours. I, too, would call the office. They will know the itinerary much better and can tell you which day might be best to visit the Vatican.
If you decide against the Vatican, I can highly recommend The Borghese Gallery. You will need to purchase tickets in advance, by appointment only. This museum has beautiful Bernini sculptures, along with works by Canova, Caravaggio, Titian, and Raphael. It is far less crowded and was truly a pleasure.
If you like to eat, and like sweets, make sure to visit the Jewish Ghetto. Pasticceria “Boccione” Limentani (Via Portico D’Ottavia,1) makes some of the best Pizza Ebraica (Jewish Pizza). This is made with candied cherries, almonds, pinenuts, and raisins: plus a few other ingredients. Think fruit cake, but without the spices and the overly sweetness. Go early, they sell out fast.