The best way to minimize weight is to use stronger rings and a larger ring diameter combined with smaller wire gauge. This minimizes the amount of scales per area which makes up the bulk of the weight.
For example if you use 18g 5/16 spring stainless instead of 16g 5/16 aluminum the rings will have a similar weight and strength but because of the springback in the steel rings the finished piece will have about 25% less weight of scales.
Another option is to use split rings, the #7F ones from worthco.com work well with large scales. I haven't used TRL's split rings in a long time but last I did they were pretty terrible.