Is it permissible to refrain from helping a woman who constantly asks for money and claims to be in need, knowing that her husband is well-off, and is it a sin to disclose her situation?

There is no blame upon you for refraining from helping this woman as long as she is not in dire need. However, you should not decide to disbelieve her or withhold compassion from her. Instead, advise her and guide her towards modesty and truthfulness. Informing your colleagues about the reality of her situation, that she is wealthy, is not a sin because it does not deprive her of anything she is entitled to. However, it is not permissible to defame her or speak about her unnecessarily, as this constitutes backbiting.

It is permissible to give her voluntary charity (Sadaqat at-Tawawwu'), as it is permissible to give it to the wealthy. As for obligatory Zakat (definition), it is not permissible to give it to her if she is not among those eligible for it. Furthermore, needy relatives have a greater right to charity.