He just mentioned it has a 4 speed manual behind it now.
But as the OP says he had the same vibration when he used a 904 auto.
@Purepony: When you replaced the 904 trans, did you find a flexplate with a half moon shaped section cut out? Those supposed to be used on the external balanced 360's.
If that was there you have a big chance indeed the vibration is only caused by the harmonic balancer, since swapping flex plate to flywheel did not solve the vibration issue.
I am not sure if you used any specific flywheel for your application though, i would assume a flywheel for a 360 also supposed to be "balanced" for external balanced engines.
From what i found on Summit is that the flywheels are used for both applications and come with a removable weight that can be used/removed as required for internal & external balanced engines.
You really need to know what is there now to get rid of this unbalance, these vibrations can be very destructive for your engine depending on the severity.