If an auditor leaves a condition in the engagement letter that if he finds errors he will want to renegotiate his fee to cover the additional work/time involved and the client is then not agreeable to the additional fees, should the auditor cease the audit work at that point or complete audit simply giving a disclaimer of opinion.
In most cases the sensible way to rpoceed would be to warn the client of an impending qualification or disclaimer. If the client refused to pay more then a qualification or disclaimer would be appropriate if the issue was material.
A resugnation does little good for the shareholders to whom the auditor reports. Better that they get some audit information than none.
I wouldn’t think that the auditor will stay for the following year’s audit however.