It's possible that I don't understand your response, because I don't see why one only needs to know about one charity. On reflection, I don't think that it is an impossible problem either. It's making uncomfortable tradeoffs - which stopped me from considering this sort of question for a long time - but I've come to believe that these questions are too important to flinch away from.

One thought experiment to consider: suppose the homeless family themselves had to choose between a free wooden house or lifesaving medical care for their child. What would most families choose? What would you choose in their place? Is there a right answer?