What is the ruling on a young Muslim man marrying a Muslim woman civilly in another country, after her family refused to marry them on the grounds that his brother had proposed to her previously, and is there another solution?

If the girl's family refuses to marry her to your older brother and thus prevents her from marrying, she must obey them, because this is obedience in what is right. Disobeying one's mother is an act of defiance and a sin. She is not permitted to travel with you or marry without her guardian's permission. If you cannot convince her family to agree, then look for someone else, for Allah enriches all from His bounty, and "Perhaps you hate a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah knows, while you know not."