I have a similar set up to Mitch, as well as a similar car. I run my 70 Challenger years of autocross, and several track days with an automatic 727 behind a stock 340. The trans has shift kit mods to it, higher stall converter than stock. I have the stock radiator cooler and and aftermarket cooler in front of the radiator, a deep pan with cooling tubes. I've run with a range of 3:55 to 4:10 gears. I should have a locking trans filler tube...on the "to do" list, as that is one point of contention. Even with the additional steps towards trans cooling the consensus is that I continually ballooned the converter, sending the crank into the thrust bearing, causing no failure, but a significant bottom end rebuild. Crank had a huge groove, and the bearing was toasted. Took some welding, new bearings, and a lot of patience to repair. Car never gave a sign of weakness and it was uncovered during a motor pull when the crank thrust was measured with a yardstick, versus a dial indicator.

Auto trans for non-straight line? Sure.
Do I like it? Sure. My car was an automatic before it was my race car, so I made do with what I had. I'm of the opinion that the automatic would do just fine. Just go big on the trans cooling, and oil cooling, and engine cooling, and...you get the idea.