I recently received a call from a woman who was very upset about her public employee union's support of very liberal candidates and causes. Why should she be forced to support the union when they support pro-abortion candidates in violation of her religious convictions, she asked. What's her recourse? Seek a religious exemption from her union dues going to the union. She has a right to have all of her union dues sent to a nonprofit organization. Not just a portion. There's been big legal battles over this right but the courts have consistently upheld the right of employees to redirect their union dues to nonprofits when they object for religious reasons to the causes supported by the union.
Here's an article on one court case and for more background information on the subject you can go to the National Right to Work Legal Defense Foundation web page. I'm told they'll even defend you for free. If enough union members, private or public, opted out for the religious exemption to redirect their union dues to a nonprofit, I think it would have an impact.