Does the elevator at St Peter's take you to a level where you can walk around the perimeter of the inside of the basilica? (Really don't want to do that claustrophobic leaning staircase to the cupola again!)
Yes.
The elevator is outside, on the right side of the basilica. You can walk part of the inside of the base of the dome, as well as the roof of the basilica, but not the entire perimeter inside.
The "up" elevator arrives in a fenced-off area that allows very little walking around the roof. You can get to a walkway around part of the perimeter inside, but there are no signs leading you there. Then you have to go to another exit from that inside walkway to get to the "down" elevator, also in a fenced-off area on the other side of the building. That elevator descends into the inside of the basilica on the far side.