Is it permissible to return with a new contract after two verbal divorces and the wife filing a Khulʿ lawsuit, or is the wife considered irrevocably divorced by three?

Divorce occurs when the husband pronounces it, and its documentation is not a condition; thus, the two divorces you pronounced have taken effect. It is not permissible for a woman to seek divorce (talaq) or annulment (khul') without a valid Shar'i reason. The original principle for khul' is that it should be by mutual consent. If a Shar'i judge rules for khul', it takes effect. Scholars have differed on whether khul' is an annulment (faskh) or a divorce (talaq). The majority of scholars hold that it is a divorce, and based on this, the wife becomes irrevocably divorced (baynunah kubra) and is not permissible for him until she marries another husband. Some scholars have held that it is an annulment (faskh), and in this case, she would still have the two previous divorces.