I think this set of links can be maintained, depending on what you believe can be the case. I think this no different than pretty much any family one tries to research in the 1600s and early 1700s. I have no original documents, but let me state what I can find online, and you decide if it makes you feel any better about it.
Let me start with: Bernard Nogaret De La Valett, 2nd Duke d'Epernonb: 1592d: 1661 Paris, Fr.(Duke de La Valette 1622-31; 2nd Duke d'Epernon 1642-61; Col. General of Infantry 1642-61; Governor of the provinces of Bourgogne and Bresse; peerage extinct 1661; dukedom sold to Pardaillian-Gondrin, Sg de Savignac.)
If you squint your eyes just right, I think you can see that "d'Epernon" can become Epperson/Apperson, and also that "de La Valette" can become Wallett. Since Bernard is the Duke de La Valette and also the Duke d'Epernon, it should, in principle, be possible to interchange the names Wallett and Apperson/Epperson.
Bernard is said to have married three times, and by his third wife had:
John d'Epernon De La Valette, b. 1638 in either the Netherlands or at sea. He is said to have died either in England, or America. This John married twice, his second wife being Elizabeth Beard (b: Abt. 1653 Plymouth, England m: 1671), and with her he had children named William, Richard, Thomas, and John.
This last John:
John EPPERSON b: 1678 England d: 1722 New Kent Co., VA
The last name became Epperson/Apperson to change it away from what would have been associated with the descendants of Bernard. I think this was pretty common due to the religious-political climate/conflict of the day.
So that is the good news. Here is the bad news:
How does this match up with contemporary records which mention these people? I have only the St. Peter's Vestry and Register to go by, and since I don't have it in front of me, I have to do this by memory. As I recall, the Vestry book identifies a John Epperson, (and this would be New Kent Co., VA). The problem is that as I recall, the Vestry book says this about a John Epperson:
John son of Jno Apperson Jur Born Feb. ye 4th X’nd Mar 21st, 1713. [p. 10]
So if this "John III" was born 1713, and John II was born 1678, and John I was born 1638, and John I was the son of Bernard, then everything is fine.
But if John II died in 1722, John III would have been only 9 years old. Wouldn't the Vestry/Register make mention of someone being given maintenance for John III? Someone would have had to have raised him.
Does this make sense? Can anyone prove any of these things I have said to be true or untrue?
Like I said, I think it will depend on what one chooses to believe.