The neat thing is that the weight loss also decreases the load on the drivetrain so it doesn't have to be so beefy.
Look at trucks... most of the time, a one-ton has the same engines as a half-ton, but the transmission and rearend are one or two grades beefier, the U-joints are twice as huge, etc. Same power, different expected loads.
You need an 8.8 to put up with a 5.0 in a fullsize, but for years Ford only ever used the 7.5 in the Mustang GT. Or look at the RX-7 guys making 300-400hp with the stock rear diffs, they don't hurt them because they can't load them heavily enough unless they dig out the slicks and abusive driving.
Keeping things under 1000lb is going to be difficult. My gut feeling is that a transverse mid-engine setup will be the lightest, since there's no rear axle, so probably that D16 drivetrain would be just the ticket. That said, supposedly an entire Swift/Metro 1-liter engine/trans combo is under 300lb... be a bit down on power, but why settle for 1000lb when you can try for 800?