I know families who have lost babies, that have had a professional photographer take pictures of the family with the baby after it died. Just so they could have more than a few pictures with the infant.
The object of funeral photography isn't to have a "posed" picture. You are catching the moment, not requesting people to smile. That I can see being awkward. Some of the most beautiful pictures that I have seen have been from a funeral. My mother in law's mother died this past summer, and she hadn't seen her mother for 15 years, and the pictures of the grave side service are all she has to remember her mother by.
There is nothing wrong with capturing a moment, even if it is at a funeral. When I photographed my father in law's mother's funeral, thats all I did. I didn't take pictures at the service due to the fact that it was in a church, and I'm not quite sure that it is cool to take pictures in a chapel, I could be wrong though. But by the grave when the family had the prayer and their last moments before the funeral was over, I was able to get some amazing photographs of the family, location and of the short graveside service. It is a memory that some people would like to remember.
I don't kow if I could ever charge for photographing a funeral, but I know that I could "shoot" one if asked to, all you have to do is treat the situation with dignity and respect.