Have you considered another Browning stainless? Brownings are usually pretty light weight. I have a Savage model 16 in .223 Rem. It will shoot lots of loads inside 1.5" at 100 yards and puts some loads inside an inch regularly. How many types of factory ammo have you tried in yours?
Another option would be to send your 16FLSS out to have it "accurized". This usually involved trueing and glass bedding the action. A couple outfits that specialize in Savage rifles are:
www.sharpshootersupply.com and
www.savagegunsmithing.com. Figure on spending a couple-hundred dollars.
Not many lefty, stainless rifles out there. Other than the Savage 16 and Browning Stainless stalker, the only one I can think of off the top of my head is the Ruger stainless sporter, and that only comes in a few long-action calibers. You could get a barrelled action from Montana rifleworks, but you'd need a stock for it, and the combo would cost around $1,500.