I had my first '56 Cadillac in 1958, through around 1956. I went through all of that ballet with the "Dump-and-fill" clutch that all of you speak of. One of the big changes was changing the 4 tangs to a splined shaft. It wasn't cheap then, so now.....hoo, boy!!
I now have my second '56 convertible, and it has a very low mileage on it, which suggests hat it has the original transmission. There were lots of changes made to that transmission, and it was never straightened out until 1958. For the '56-57 models, the best controlled coupling transmission will be a '58 transmission, with a '56 tail housing and shaft on it. That gives you the rear pump for push starts, which was dropped in 1959.
In one of the Cadillac parts books (1961, I believe), Cadillac had a replacement transmission shown for the 1956 and 1957 models. It was a totally new transmission that could be bought, and completely replaced the original transmission.
One of our members in Texas has also installed a Turbo Hydramatic 400 transmission in his '60 Cadillac. I've toyed with that idea for my current '56.