Couselling will be CONFIDENTIAL, as far as I'm aware, so they might know you're going to therapy or counselling but they won't know anything more than that. But surely as far as they're concerned, if you are IN counselling that is a positive thing because it means you're managing your problems?
I'm in the UK but I assume it's the same, therapy is confidential, they won't discuss anything with anyone else UNLESS they feel there is an actual risk to you or someone else, and that isn't the case with you so it would be confidential. Maybe try ringing up and asking about their confidentiality, express your concern and ask if it would be suitable for you. Maybe you could book one session to discuss all of that, and then if you are ressurred you could have ongoing therapy? It is probably best for you and your daughter, and they won't take her off you unless there is a danger to her, and there isn't.

Maybe call a helpline as well, or instead, to ask for advice and info on this all.