How to present preprints that are not publicly accessible on a CV for math academic job search?
If the paper has been submitted, then of course you can list it as "submitted", whether it's public or not.

Otherwise, I think that about all you can do is to say "in preparation", but this won't really have much impact.  My impression is that "in preparation" means about as much to a search committee as "I had this idea for a paper", i.e. practically nothing.  They aren't really likely to believe a candidate's estimate that the paper is "99% done", since it's obviously self-serving.  And anyway, the last 1% is often where you discover a critical gap that takes months to fix, or can't be fixed at all.

If the paper really is done or nearly done, your best bet is to work hard and try to get it submitted or posted publicly (preferably both) before you send in your job applications.

If you can't get to that point, then the next best thing is to make sure that at least one of your recommendation writers is familiar with the project.  (If you have co-authors, ask one of them to write you a letter.)  They can write about how awesome it is and assure the committee that it really is practically done.  In this case you still say "in preparation" on the CV, but you have some third-party backup.