Is it permissible for a man to take back his ex-wife after an irrevocable divorce (طلاق بائن) in exchange for a debt owed to her father, knowing that the divorce was a single irrevocable divorce (طلقة واحدة بائنة)? And does this require a muhallil or is a new marriage contract sufficient?

Khul' (divorce at the wife's instance) expressed with the wording of talaq (definition) (divorce) is considered an irreversible divorce (talaq ba'in) and counts as one of the three divorces. This is the opinion of the majority of scholars. If this is not the third divorce, the husband may return to his wife with a new marriage contract, in the presence of her guardian (wali), two just witnesses, and a valid صيغة عقد (marriage formula). This does not depend on her marrying another man.