Senators also advance Nicole Berner's nomination
Challenges come despite bipartisan interest in addressing the problem
Calif. Democrat elected to U.S. Senate in 1992
Three Republicans voted for Rita Lin's nomination
Bill received final approval last week
Bill passed with bipartisan support
Bill approved by 61-36 vote margin
Anti-LGBTQ groups have launched conservative effort to block bill
Bill advanced this week with Republican support
First vote on bill expected on Wednesday