Why does the King of France accept Cordelia when Burgundy rejects her?

France and Burgundy are, as Lear explains, "rivals" in Cordelia's love and have been waiting in "amorous sojourn" in Lear's court for some time. Lear knows that both of these suitors are interested in whatever dowry Cordelia will bring to the potential marriage, so when he has disowned Cordelia, he declares, "let pride . . . marry her," believing that both her suitors will now undoubtedly withdraw their suits.
He is right about Burgundy, but the King of France questions Lear's judgement by saying that it is odd that Cordelia, who was once "most best, most dearest" to Lear, should now be so utterly reviled by him. He says he does not believe that Cordelia could ever have committed an offense of "such unnatural degree," given what he knows of her. Upon hearing what she has done, he says, "is it but this?" and declares that Cordelia "is herself a dowry."
Burgundy asks Lear to provide "that portion which yourself proposed" and then he will marry Cordelia, but when Lear refuses, he withdraws; he will not take Cordelia without a dowry. France, however, says that Cordelia is "most rich, being poor" and that he loves her for her virtues; nobody will take "this unprized precious maid" from his grip, so impressed is he by her principles and so offended at Lear's treatment of her.