IIRC some men DID sue for breach of promise. but they were problaby considered ungentlemanly to have done so.

I think that the idea that only a woman could break off an engagement was to do with the idea that a woman's career was marriage..If she was engaged to a man, she made plans, had expectations of being kept and looked after financially for her life and having given her promise to one man, she could not accept another suitor. She also might have spent money on her trousseau etc.
So if the man let her down, she had maybe sacrificed a few years of her life and her "nubile years" in vain. So she was entitled to soem kind of monetary compensation and alos it was "ungentlemanly" to let her down unless her fiance had discovered some serious fault in her. That's why a scrupulous good hearted young man like Edward woudln't do such a thing to Lucy.. He knew that he did not love her, that she was silly and vulgar and uneducated and not a suitable wife for him but he beleived that she was still attached to him and wouldn't let her down. Esp no doubt knowing that she was poor and might not get another suitor. So that means he's stuck with Lucy unless she chooses to break it off.