Will this 904 trans work in 65' Barracuda?

As long as you have a single cable floor shifter or don't mind putting one in that trans. will work. a 3 cable shifter wont' work.

A 360 is an externally balanced engine like a Chevy 400 small block but where-as Chevy used a special weighted flex plate to balance the 400 Mopar used a weight on the torque converter to balance the 360. But the aftermarket has came through with a special flexplate that takes the place of needing a specially weighted torque converter. Search Summit for B&M or TCI flex plate that'll work on a 360 LA assuming your engine is a 91 or older or 360 magnum if it's a 92 or newer. The aftermarket flex plate I'm talking about will have a big notch cut out of it on one side. That's what does the balancing.