To cut to the chase, Lawrence is Sarah's lover. He also happens to work for the Home Office, the department of the British Government that handles immigration. He is married to Linda and has three kids. He's known for his self-deprecation – constantly calling himself a loser – which Sarah finds alternately annoying and endearing. Otherwise, Lawrence isn't very sympathetic. He calls Little Bee a b---h and threatens her, and might have consciously manipulated Little Bee into exposing herself to the authorities when Charlie (who was just hiding) was thought to be missing by the River Thames.
In the end, he does play the good guy (sort of) and use his position at the Home Office to find out which plane Little Bee is being deported on. Giving Sarah this knowledge is a big sacrifice on his part. He knows that it means that Sarah will be with Little Bee in Nigeria, not in England with him. At the same time, his act is what makes any future relationship with Sarah possible, and, in spite of his faults, he does seem to love her.
Lawrence might also seem unsympathetic because he's a lot like many of us. We don't let strangers into our homes. We don't seriously consider upsetting our comforts by sponsoring refugees without documentation. We probably wouldn't be brave enough to cut off a finger for a stranger. Lawrence, like most people, isn't willing to start making sacrifices that could disrupt his life, even if he'll admit it's a bit unsatisfying.
Unlike Andrew, Lawrence is able to deal with the reality of inequality in the world just fine. He doesn't lose any sleep over it. But, he can't be with Sarah and hang on to that apathy at the same time. Making sure Little Bee is safe and sound, and helping make public the situations faced by others like Little Bee, becomes Sarah's life work. In spite of their rather tawdry beginnings, Lawrence seems to be turning into a nobler person as a result of knowing Sarah.