WI: French victory at Poitiers (1356)

On the topic of HYW battles going the wrong way for the English (i have made previous threads on Crécy and Agincourt)...
Suppose that John II of France and his army manages to defeat and rout the English at the Battle of Poitiers, in 1356 AD. Either that, or the battle is avoided through negotiations.
Now that the English have been more successfully removed from Aquitaine, what happens to France then? John II and his court are free, so can they negotiate for more lenient terms at the Treaty of Bretigny?
Assuming that Edward, The Black Prince, perishes in the battle before leaving any issue, what are the effects on English court politics from there on in? If he doesn't perish, will he decide to refocus his political efforts towards England proper?
 
The removal of the Black Prince makes Lionel of Clarence Edward's heir. This will have interesting consequences if he still has no son.
 
Who would be the most likely candidates for the throne if and after Lionel dies with no male issue?
His daughter's husband would have a jure uxoris claim, especially if they have male issue, though he may settle for regent. Next up would be Gaunt and his line, I suspect he would have strongly pushed for a declaration of heir presumptive on his line and also marriage between his son and Lionel's daughter.
 
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