Yep wrong flex plate 383/400 needs balanced flexplate. If you used std trans setup for 350 its the wrong one. Call the guy you got the kit from and tell him what happend. they should be able to provide the correct part number. Eagle has the correct part on the shelf but its got to come apart. You can also call them and tell them what parts you used instead of the kit parts. They do make one for std trans. Just got to get those parts. i think its just a lump of metal you can bolt on. But not sure if that is a proper way to fix the issue.
Also a shop balancing the engine is the correct way to go. Balanced kits can usally just be run but it is best to balance it for high rpm running. This is not the same type of balance that is on the flywheel or harmonic balancer. Lots of info out there. You will have to dig around the net to find out what is right for you. Look up "383 internal or external balancing" in google.
Do not drive it by the way. That can damage it further. If you have to drive it keep the rpm below 2K for a short trip. I would not drive it sep into the garage. Sorry i know it sucks to have to pull it back apart again. But you may be able to just drop the trans off the mounts and install the new setup.
Hope this helps!