]Edward was the one who offered to let her out the engagement because he had lost his inheritance and family connections and felt it his duty to give her the chance to change her mind

Ch. 38: "He could not bear to think of her doing no better, and so he begged, if she had the least mind for it, to put an end to the matter directly.."

Maybe Edward hoped that one of the side-effects of standing his ground and doing his duty was to get Lucy to drop the engagement. It sounds to me as if he is saying "I have no money, no connections, no prosects - do you really want to marry me? Really? Still? But you deserve better so please just go if you want to. I won't hold you back"