Yes, the copper band and the steel sleeve both go on from the bottom.
The projectile is held in the chuck at the fuze end with the body on a steady rest, or center drill the base and hold it with a live center. Or both. For a 5" you’d probably need both.
I’ve never tried to make a band by rolling sheet copper. You’d have to figure a way to hide the seam and layers. And, would copper expand enough to heat shrink it? A 5"/38 band would have to be about 5 1/4" in diameter to give you enough material to turn it to shape. The inside diameter would have to be whatever diameter the score cut is.
I’ve heard of guys making a band seperate from the projectile, cutting it in two pieces and them gluing or brazing it onto the projectile. And then turning it to final shape. If well done, the joints should be barely visible. That would be my second choice for something as big as 5".
A big enough piece of copper can sometimes be found in a junk yard, or more likely at a recycling place, since it is valuable. You can buy new material from a metals dealer but I don’t know what you’d have to pay for a piece big enough for a 5 inch. When the cost gets that high it’s time to start looking for a projectile with the original rotating band still in place and do repair work at the pointy end. Here’s a 5"/54 that I rebuilt that way.