Is it permissible for one entrusted with donation funds to give a portion of them to a needy female relative who is too proud to accept these funds, as a gift without informing her, and would he be sinful if she found out and refused?

It is permissible for the giver of charity or their representative to give voluntary charity (sadaqat al-tatawwu') to relatives and non-relatives alike. There is no blame upon your relative if she refuses the charity if she is not in need of it, in order to avoid feeling indebted. Scholars have not explicitly stated a ruling on giving voluntary charity to someone who does not accept it without their knowledge, nor whether there is sin in doing so. If your relative returns the charity, there is no blame on either of you, and you should give it to another person in need.