I have sent an academic job application with a cover letter starting "Dear Madam or Sir,..." and I've just realised that the correct form is instead "Dear Search Committee Chair,...". Have I made a huge mistake?
Probably not. I doubt anyone is going to catch it and, if they do, they probably won't care.
You haven't even made a small mistake. The salutation you used is just fine. I actually like it better than the other idea you find (Search Committee Chair).