I'm getting into material culture a little here, but it's in order to better understand popular entertainments. Given that we know something of how historical games in the jeu de paume family were played, what sort of balls were used?
There are two kinds of balls of interest that are still used to this day in modern Valencian pilota games: the pilota de badana, used in the street game of llargues, and the pilota de vaqueta, used in the higher-status raspall. These balls very probably match two of the three types of tennis balls described in Vives' dialogue.
Vives' "French" tennis ball is the easiest to comment on - there is one that survives from 16th century England, and it matches the description in the dialogue precisely. Today it is in the Museum of London collection, and we can see from the image on their website that it appears to be made of eight approximately-triangular sections, stiched together in such a way that the stitches do not show from the outside. The ball is made of white leather, and is stuffed with dog hair, exactly as described by Vives. It is a near-perfect match for the modern Valencian pilota de badana, which is constructed the same way; the Valencian balls are black to contrast with the white walls of a trinquet (a Valencian handball court), but they are made of 16oz leather in the same shape, and again, stitched in such a way that neither stitching nor stitch holes are visible from the exterior. Modernly, they are usually stuffed with a synthetic batting rather than the dog hair used in historical examples. They are then pressed into a mold to make them properly spherical rather than merely approximately so. (A video of the manufacturing process can be seen here.)
In contrast to the French ball, Vives describes Spanish balls as being made of thinner leather, of unspecified color, and stuffed with rags. Fabric-stuffed balls were quite common historically, and survive today in the form of the Valencian pilota de badana. While I have not found information about the pattern used for a historical fabric-stuffed ball (I'm sure it's out there!), a modern pilota de badana is stitched in the same pattern as a baseball, and is typically comparable in size. I am not currently aware of any surviving fabric-stuffed balls for comparison, unfortunately.
When I make balls for historical handball, I prefer a slightly different design, which I admit has no historical basis (it's inspired by balls found in early medieval sites from eastern Europe, but I have no reason to believe it was ever used in the west for games in the jeu de paume family): two circles and a rectangle, forming a cylinder. The rectangular piece should have an aspect ratio of about 5.5:1, and should have a length ever so slightly greater than the circumference of the circles. A slit is cut along the length of about the middle quarter of this; the pieces are then stitched together. (I like to use a running stitch, right sides together, then invert the ball and whipstitch to reinforce the seams.) Next, scraps of cloth are stuffed into the ball through the slit, which is finally sewn shut with a whip stitch. I've found the base of a standard American pint glass makes a good template for the circles; this gives a size somewhat larger than a pilota de vaqueta, which seems a reasonable guess. I use this pattern because it is easy and produces a thoroughly serviceable result; if I am able to find out what the historically correct pattern would have been, I will switch to that in the future.
A fabric-stuffed ball has some bounce, especially if it is packed as hard as you can manage. It still won't bounce as much as a modern lawn tennis ball, of course - which is a good thing, if you're playing outdoors and don't want to lose your handmade leather balls. (I presume the vaqueta-like Anglo-French balls were bouncier; their modern counterparts bounce more than the modern pilota de badana does.)
Lastly, there are passing references in Vives to larger balls, filled with wind, which must be struck with the fist. I strongly suspect these would have been made from inflated pigs' bladders in leather cassings, like early footballs, and that the game as played with them was essentially an early modern form of volleyball.