If a woman has been irrevocably divorced by her husband with a major divorce (baynunah kubra), and she lives in a country where she currently cannot access a Sharia court to request khul', is it permissible for her to marry another man before obtaining the khul' document, and would her marriage be invalid or would she be sinful if she did so?

A wife cannot consider herself irrevocably divorced by judicial decree, for not everything established by a fatwa (definition) is established by a judicial ruling. The judiciary rules based on outward appearances and evidence, and the husband denies explicit divorce, and she has no evidence for an implicit divorce. Even if the husband were to admit it, he could excuse himself by claiming forgetfulness or similar reasons. Therefore, the wife has no option but to await a judge's ruling for divorce or to seek khul' from her husband. Her marriage is not permissible before that.