I'm one of those unlucky few that have never gotten that proposal, so I have no idea. I would guess that you would keep pumping friendly interactions and keep your finger crossed.
However, if you're a gold-digger, then you should be fully able to marry the other and in a marriage, you can choose between both houses. That will change your last name, but you can change that household's name in 'Edit Town' and the sims' names in the courthouse, so this is the usual route I take.