I don't actually know the correct id numbers for SN 4063, but we may be able to figure out if the parts are what should have been in that GTE. First of all, the numbers you gave us for the Gearbox and rear axle are not the ones we are looking for. The number you gave for the gearbox is the manufacturers number and it was poured with the case. It is identical to the number on my Series III GTE (# 4247), and I would assume all GTE's of that vintage. The number you gave on the gearbox is similar to mine, but not identical.
The number we really need was stamped on by the Ferrari factory and was unique to each car. On the axle, it is to the left of the drain plug, as viewed from the rear of the car. I have tried to attach a photo of mine to show you what I mean. It is my understanding that the number 66/63 means that it was the 66th axle used in 1963.
The Gearbox number is on the top of the gearbox, on top of a protrusion on the driver's side. If you cannot remove the leather transmission cover to read it, you will have to stick a mirror up from under the car and fish around. That is, unless it is like my car and completely covered in oil, grease and dirt. It should be close to the number of the axle, but not necessarily the same.
Once you have these numbers, get them to me and I will ask around to see if anyone recognizes them. The telio guys seem to have lots of sources for this stuff. I also know that Ed Niles used to own 4063 (for parts) when it had a Cadillac engine. By the way, according to the GTE Register, the "body, interior drivetrain and possibly transmission" was for sale in 1990. Doesn't say if it was a Cadillac or Ferrari drivetrain though.
You also may want to contact Ferrari with those numbers and see if they can be of any help. They used to be quite helpful, that is before they had a program where they could charge for this info. Once I had all my numbers, they sent me a PDF of the build sheet. It confirmed that my car is original and kind of cool to look at.
Where are you located? I know we all would like to see the car.