@patrat84 I can totally relate. In fact, when I travel with my kids, the huge stroller becomes our bell cart until we hit the door of the plane. I personally have never had an issue with leaving my large and heavy stroller on the jet bridge, but to be safe you can contact SWA via Twitter through a DM.
Good luck, and remember do to take a deep breath. I’m told that one day we will look back at these times and miss them. 😀