For many years, there were a lot of keis available under 800-900 lbs., but they've been getting heavier at each model change (one I was going to cite as an example is the Mitsubishi Minica, but checking it's up to almost 1300 lbs now). The Suzuki Twin you mentioned went out of production a few years ago and was a bit bigger.
What I find interesting is that cars have been getting heavier in Japan due to rising interest in SUVs/crossovers and added luxury/safety features, but in 2009 sales of keis have shot up over 10% while almost every other sector has fallen. Cost, efficiency, licensing (by weight and displacement), and parking. With that, there are promises of some even smaller and cheaper keis again...though I'm not sure they'll be as low as 700 lbs.