If you check a bag, a long umbrella (yes - they will make you check those also), "borrow" a wheelchair, or get an audio-guide, you must return these or head back towards the Museum entrance area... so No, you will not be able to use the "group" exit and head over to St. Peter's Basilica if any of these situations apply. Many tour guides will "gather" their headphones while in the Sistine Chapel which allows folks to decide if they want to cut through the Basilica or head back into the Museum.
Usually a smaller backpack will be okay. But if it looks "BIG" you will be redirected by the folks at the X-Ray machines to head to the "coat check" area. If I was carrying a backpack I'd enter the doors and head to the busiest X-RAY as far left as I could go - that's the furthest AWAY from the cloakroom. There you might get by, or could "shuffle" into the crowds as you leave the X-Ray unit... I've seen kids do this a few times.
A good picture of the staircase leading down from the Sistine Chapel to the Basilica is located at the bottom of this post: Vatican Line Info .