we stay off site as well, and we generally have dd nap in her stroller while we browse, but we do shorter days, either arriving early and leaving round 3, or arriving after lunch and staying out late.
we have always used which ever grocery store was closest, last time it was a Winn Dixie, and it was fine.
and I would take the double stroller - if you are offsite then you must have a car, so it will be no big deal. I would recommend forgetting the tram and walking to the gate, 90% of the time it's as fast as taking the tram anyway, and it's a lot less hassle then folding the stroller, we do this with our single stroller.
Last year, we had a single stroller and a 3 month old and a just 6 year old. The 6 year old walked virtually all the time, but late at night we put the baby in a sling and the 6 year old had a ride - this might be another option for you. After the
MVMCP this worked brilliantly, the 6 year old would not have made it back to the carpark on foot, but was able to fall fast asleep in the stroller while we waited for the ferry.
Bev