How to apply for a faculty position elsewhere without revealing the conflict with current dept chair?
Good search committee members are aware that many applicants do not want their current department to know that they are applying for a job. Search committee members should only be contacting your listed references. That said there are plenty of bad search committee members and plenty of search committees that accidentally let information leak out.

It is perfectly reasonable to not list your current chair, or anyone from your current department, as a reference. You should, however, list your current position on your CV. Not listing your chair as a reference does not mean your chair will not get an informal phone call, but there is nothing you can do about that. You should also know that when a department chair gives a reference, they often highlight the good aspects, and ignore the bad aspects, of people they want to get rid of and focus on the bad aspects of people they want to keep.