Is the mere offer and acceptance sufficient to enforce an un-documented marriage, or is the guardian's consent and understanding of the matter a condition, especially if the guardian considers it an engagement and denies his approval of the contract?

The best approach in matters of dispute is to refer to the Sharia court, or its equivalent, because the judge is more qualified to examine the matter, and his ruling is binding. Jurists did not differentiate in marriage between a serious and a jesting person; both their marriages proceed. They cite the hadith: "There are three things whose seriousness is serious and whose jesting is serious: marriage, divorce, and رجعة (return to marriage after a revocable divorce)." Sheikh al-Islam Ibn Taymiyyah explained this by stating that the jesting person utters the words without intending to commit to their ruling, but the rulings are tied to their causes by the Lawgiver, not by the contracting party. So, if one brings about the cause, its ruling becomes binding upon them.