If the department put you on probation and did not dismiss you outright after getting three C’s in your first semester, it’s on them. It’s their administrative error. It’s one thing to have three C’s spread across years which can easily go undetected, but you got them all at once. And they said you could continue, albeit on probation. I believe you have a strong legal case here.
Are you currently a research assistant for your advisor, and does your advisor have the funds and plan to continue supporting you until graduation? If ‘yes’ to both, your advisor could fight to keep you, all the way up to the VP for Research/Provost level. Your advisor can cite the administrative error and say how critical you are in helping to deliver on an externally funded research grant. Assuming that they have a good impression of you and your work, your committee members could also go bat for you. (Departmental politics can be complex so they may not want to stick their neck out for you.)
If your relationships with your advisor and/or committee members are deteriorating, or if your advisor’s money is running out, then you might be in a precarious position, in which your three C’s is merely an excuse for them to get rid of you now that you guys are not in good terms or the project is ending or the money is drying up. It’s very bad for you if that’s the case (I hope not) and you may have little choice but to seek a legal alternative which will be expensive and unproductive. Or go away quietly which will be hard to swallow.