Transmission Recommended to Run Behind Built 360ci Engine in a 63 Valiant

From what I gather after reading quite a few threads, I will not be able to use the 904 that is currently in my 63 Valiant if I plan to run a built 360 V-8. To get an idea of the kind of motor I will be building for this car, more details are in this thread: http://www.forabodiesonly.com/mopar/showthread.php?t=198641

I assume I will need a well built 727 to put behind the motor, but what does one need to do a 727 (Or whatever transmission is recommended) so it will handle the abuse. Also, I am curious if there is anything that I could do so the push button shifter on the dash could still be functional. It's not required, but would be nice.

Thanks,
Rob