Tire rack has Yokohama A032R DOT legal race tires in 175 60 13. That is fairly small, but may be a tight fit on 4" rims. They recommend a 5" to 6" rim for it. Kuhmo also makes a DOT legal race tire in that size, the V700. I think Avon does too, but nobody seems to be selling them here. I have a few sets of 13" autocross tires, a couple are 185-60-13 Yokohama 008 RSII's and a set of Hoosier Autocrossers (non-radial). They are not as sticky as new, but way way stickier than ANY non race DOT legal tire. I drive my Lotus 7 on Hoosier V700 Victorracer DOT race rubber all the time. They will get hard long before they wear out. And they will still be grippier than high performance tires. I'm going to buy another set of the Hoosiers to replace the Yoko's mounted up for autocrossing the Cortina.
I would call Tire Rack if I were you.
Whoops, I thought that you were looking for something sportier than so called "high performance" tires.
I use race rubber on the 7 all the time, and it works fine. I tried it on the Cortina, but it didn't work well on the street. It was way too nervous, and followed road irregularities extremely badly.
A spridget is probably about half way between the 1100lb Seven and the 2000lb Cortina. It might drive great on race rubber.