You are actually asking two questions. I believe that you are interested in open-track road courses and/or local autocross events (Solo2).
As far as SCCA classes, it would be very difficult to get your car to pass tech in the generic "run anything" ITB class. Raise the roll bar, update the belts and fuel cell, install Lexan, mount fuel inspection port in the fuel line, and more; get Nomex undies, SP 5-3 or better driving suit, and some arm restraints for the cam-lock; then, take a school and run a Regional or Double National event would be the start.
NASA's time trials sound cool and you "run whatcha brung". Check your region's website to determine qualifications. Take the penalties for weight and C.I. displacement, and just go have fun.
SCCA's Solo2 is commonplace in most cities. Your replica would fit easily in the AP class (if your group deleted that class last year, then you will run XP). If entries are low, then they bump you into the nearest class that can field cars. AP bumps to BP bumps to CP bumps to DP... basically including heavier and heavier cars as the class shifts.
Here's the real kicker: Solo1. Same classes, same simple tech rules ("secure everything and snap on an extra throttle spring"), but the speeds go from 40 mph maximum with the course run entirely in 2nd gear (Solo2) to 75 or 80 mph max with the driver sawing the wheel and running through the gears. Tho Solo1 eats up tires and brakes in a few runs, I think it is absolutely a blast. Ask around regarding the high-speed Solo events. People will tell you which events to sign up for. There is always a pro that conducts a novice walk-thru early in the AM, so don't let the course intimidate you. Have fun and remember that you will need to work out on the course for one of the Heats (or volunteer in Tech and your work assignment is out of the way before the driver's meeting--cool!)
Wish you the best,
Paul
SCCA TX Region
Note: In this event I ran AP against two Superformance 427s. I got the trophy at the end of the day, but they had more fun! I normally run BP (a vintage Vette and Cobra 427s when AP gets bumped up). Cobras with heat-cycled sticky-compound tires are VERY VERY competitive in autocrosses.
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