Man that's tough! There are already kits for both for other cars, but a lot will depend on how much room there is around the engine on the Slingshot. We would have to see under the hood to figure out what would be the easiest.
Turbo all the way! Why rob power to make power? Just use the exhaust gasses to make all the power. If this thing is super light, the power of the motor will be more than enough to get it off the line, so lag will not be an issue.
I guess we will find out how light it actually will be soon. I would guess without a top, no doors and down one wheel and tire that it should be under 2,000 lbs. At least lets hope it's in that ballpark
If supercharging a intercooler is a must on the twin screws. They get heat soaked fast. I'm not sure yet what rout I'll go yet since I haven't driven one yet. But I am leaning towards a turbo. Seeing some of the reviews and vids its pretty easy to light up the rear tire. A SC make almost instant tq and power down low so traction could be tricky. With a turbo I can add a aftermarket ecu and boost by gear to help with slip and use more of the power. Biggest concern is how much HP will that belt handle before it snaps!
I had to go research the difference. From what I read...it sounds like a supercharger is less complicated and shouldn't require a cooler. Turbo should produce more power, but it's more complicated and requires cooling. A friend had a turbo on his car and I remember it causing mild whiplash when in came in and out. That was 15 years ago, but I remember hating his car.
Turbo's 15 years ago where not very efficient and had allot of lag. Modern turbos have almost no lag. A supercharger does require a cooler if you want to make HP or if you don't want to worry about heat soak. Turbo's are the way to go.