I think an effective strategy would be to find a representative who is sympathetic to the cause and catch their ear. It would be especially good if they're up for re-election and their constituents favor your position. They don't have to even be in your district or state, but it would be better if they were.
In this case you want to talk to those who are sympathetic to LGBT as well as those who may be supportive of healthcare reform. Also not everyone sympathetic to LGBT would care about trans issues as much as they care about, say, same sex marriage, but it's worth a shot.
The President would also be a good person to contact because healthcare reform is his baby.