In Matthew 19, does the "exception clause" allow divorce and remarriage?

No. The exception clause never permits remarriage. Divorce and remarriage while your spouse lives is adultery, because no man can dissolve what God has joined. Jesus says, "Whoever divorces his wife, except for fornication, and marries another, commits adultery" (Matthew 19:9). The question is which clause the exception attaches to. Jesus places it between "divorces his wife" and "marries another." In Greek that placement ties it to the divorce, not the remarriage. Had He meant both, He would have put the exception after both clauses. Fornication by a spouse severs the one-flesh relationship. From the beginning, the natural law recognized that the innocent spouse may separate, and separating in that case is no sin. That is as far as the exception goes. Remarriage is still adultery. Matthew 5:32 adds, "whoever marries a divorced woman commits adultery," with no exception. Mark 10:11-12, Luke 16:18, and 1 Corinthians 7:10-11 give no exception either. Consider the alternative. If the clause allowed remarriage, the fornicating wife would be free to marry the man she sinned with, while a wife put away for no fault could not remarry. Moses allowed divorce only because of the hardness of their hearts (Matthew 19:8). Jesus restores what God intended: what God has joined, no man puts asunder.