which multi-speed hub should I upgrade my trek district to
I would upgrade to at least a 5 speed.  3 speeds have been around for a hundred years now and it is time for you to have a technology upgrade.  Sturmey AW had production problems with the move to Taiwan but they may have worked that out.  5 and 7 speeds are mainstream.  That should be more than enough to get you over the Williamsburg, Manhattan, or Brooklyn Bridges or up any hill in Central Park, Harlem or the Bronx.

More gears that that Higher than that is more esoteric.  More LBS can service the Shimanos than the S-AWs.  I would go with what your LBS recommends unless you are comfortable repairing these in the unlikely event that something goes wrong.  My favorite shop is the one on Grand Street not far from the Williamsburg Bridge.  He's been there forever.

Then if you want to drop serious coin there's the Rohloff Speedhub 14 speed, but that's out of most people's league.

Sheldon Brown (RIP) has excellent resources that are dated but still relevant.  I'd search for those.  But mostly I'd stop reading your computer and just ride.