If Kevan, the more reasonable and moderate of the two brothers, wouldn’t go near Jeyne because of her “doubtful blood,” why the hell would Tywin ever have?
As for Tywin and his bannermen, I think it’s a bit of wounded pride – Tywin was not content for his children to marry any less than the very best – and not wanting the embarrassment and loss of face of being denied by one of his own bannermen.
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