Vatican and St. Peter's in mid-September

September is one of the most popular months for travel to Italy and hotels and apartments tend to be at their highest prices. I would expect the Vatican to be very very crowded; you wisely have timed tickets but expect it to be shoulder to shoulder inside. I would take my leisure with the collection and not rush to the Sistine chapel where it will be absolutely shoulder to shoulder.

My son & I went on an early 7:30 am, supposedly "before" the crowds tour of the Vatican/Sistine Chapel. There were 100's of early visitors there & it took us easily 1/2 hour to enter. (The wondrous result of our truly early visit was that we spent about 30 minutes "alone" with no more than 40 others in the Sistine Chapel !! To me, as a religious Catholic & a Catholic educator, this was close to Heaven.)
Later in the exhausting 3 plus hour tour thru part of the museum, walking thru the Chapel again to the Raphael rooms, etc. we encountered 100's, even 1000's of other visitors.

I suggest you get to the lines well before your timed entry & I wouldn't be surprised if the gates don't magically open at 3:30. There will be 1000's there inside & out.

A seasoned Euro traveler, I am amused by your comment that, in Roma, you "would have nothing to do between the visits." It's Roma! Be in the moment and you'll realize there is way way too much to do & never enough time to do it. (I just applied for a RS Roma tour as there are Caravaggios I must see. Even tho I've visited 20 Euro countries & had terrific experiences in all, no number of Roma trips will be enough.) Godspeed. ab