With this restriction, I'm not sure some of the bikes mentioned are a good idea -well at least probably not without changing the handlebars and stem. Given what you posted about your back, I'd say consider a hybrid or a comfort bike, probably leaning more to a good quality hybrid. The best thing you can do is try it for yourself.

Another thing to consider might be to get a trailer, but that might not be practical if you can't leave it outside the store without fear of it being stolen. It would however, give you potentially more choices of a lightweight bike while still providing touring capability.

Thinking about it, I can't help wondering if the solution is a two bike solution; a cheap beater bike (or your current bike?) for that grocery shopping trip, and then a nicer lightweight bike for pleasure tripping/touring/commuting.