However, I suspect that Henry VII, whose crown still rests lightly, would be wary of a Spanish name such as Ferdinand. Isabella is plausible (it was the English version of Old Spanish Ysabel, after all), but I suspect the son would end up as Edward, Arthur, or even Jasper.
The boy would definitely not have been named Edward or Arthur. Henry VII would not name his son after the father of his dead first wife (and Catherine would not allow him). Also, Edward being a Yorkist name would very ironic for the prince considering he would not have one drop of Yorkist blood in him.
And Arthur is probably the second to last name Henry and Catherine would want to name their child (after Richard of course). It just brings back too many sad memories for both of them.
Out of the three names you mentioned, Jasper is the most plausible (Jasper Tudor would probably be still alive at the time of thr boy's birth). However, Jasper is not a Christian name and not even a European name. There was no Saint Jasper and I can't think of any European monarch during that time that didn't share their name with a saint. Now keep in mind that when Jasper Tudor was born, he wasn't royalty even though his mother was a princess because she married a commoner. Therefore, there was no real requirement for him to have a Christian name.
This leaves us at Ferdinand which is probably what Henry and Catherine would've named their son. Ferdinand II the boy's grandfather would've liked it and this would very slightly strengthen his relationship with Henry, something Henry very much needed.
If Henry and Catherine had a second son they would probably name him John after both of Catherine's grandfathers, her dead brother, John Beaufort and John of Gaunt.
Alright, enough about names.
Prince Ferdinand would not be a link between the Houses of Lancaster and York and instead be only a Lancastrian. As a result, would it be disrespectful for his father to make him Duke of York? Maybe instead he would be the Duke of Clarence or something else?
Also, is there a possibility of a Second War of the Roses this time with Henry VIII representing the Yorkists and Ferdinand representing the Lancastrians? How could this happen? If this happens would Spain and the HRE join the war on the side of the Lancastrians? Can religion also possibly play into the conflict with the new Yorkists as Protestants and the new Lancastrians as Catholics?