It's very sad that your T and his family are experiencing such a terrible situation. I do hope she is okay.
I do want to say this though - a therapist has a duty to ensure they are fit to practice. I'm sure he will have made an assessment of his own situation and emotional needs, and clearly he has decided he is able to take you on as a client at this time. That means he deems himself okay to deal with your billing. That's his call, and while I think you are kind to be so considerate of his situation, you also need to trust him to deal with his stuff himself. You deserve that.