Good advice on here. I want to second a couple of things.

I also have two cars, one is a street 68 Road Runner with a 4 speed. I would never change this over because I do not race it and enjoy shifting on the street.
My second car is a 1970 Plymouth, (Was a Satellite but it is in transition to a Superbird, so for now will will just call it confused). This car I bracket race and run 7.0 index class with. I absolutely would suggest you do a Cheetah, (Turbo Action, like Cab said) reverse valve body in the transmission, along with their shifter. The shifter is not a cool or pretty as some but it is reliable and very robust. I burned up several shifter cables with my headers, and once I put the Cheetah on, problem solved. Oh it still burned the rubber off the cable eventually, but the wire wrapped housing is so strong, it has never failed or even needed to be readjusted.

Can you race a 4 speed, yes, but to be competitive, you have to either race in a class like gear jammers. Go to the track and watch. I knew Bullet Bob Reid back in the day. He had a record setting 68 Barracuda, and always shifted gears. He had the best set ups money could buy, but still often times broke. If you race, you will push it, and eventually it will likely break. In my opinion there is more auto trans parts for racing, since 95% of the folks racing use them.

Bottom line is you will need to decide between the fun of running a 4 speed vs the consistency and reliability of the manual valve body automatic. Good luck and have fun. This is all about the journey, there is never a final destination to working on these old cars.