Check with these guys too.
https://www.foureverfour.com/
I doubt you'll find any actual history that your car raced in the 30s. A car with proven history would be snapped up at a premium by someone wanting to vintage race it. Most of the vintage race groups require cars with a proven pedigree.
If your vehicle dept wants the car to have actual "old" history just search the internet for an old B&W photo of an A racer and declare your car is the one in the picture.
The mechanical brakes would be fine on a speedster. They are adequate on a normal car and even better when you take a few hundred pounds off the car.