@Beth Twigger, very interesting question. I work for a CS Software company myself and we often get this question. In summary, our response is that A)you have control over the data you choose to pass, so your CS software provider should work to accommodate this scenario and/or most organizations like yours choose to pass data to their CS software may not be considered pii (membership type, customer journey stage, or ARR as examples).
However, I recognize this is an extremely sensitive topic and the answer is dependent upon you own unique situation. Happy to chat more if you would like OR put you touch with some companies I know that are using a CS solution and whose primary customers are healthcare/financial institutions.