Didn't see that she can't visit to Canada earlier. Because then the solution was a simple civil ceremony in Canada that is cheap and gives you a certificate. Is it possible to visit a 3rd country and have a civil ceremony and register your marriage? You mentioned Lebanon recognizes your marriage, why not first try with the Lebanese consulate first, and if not that, visit Lebanon or any third country and get a civil ceremony done? However make sure, you would need photographs of your marriage ceremony and proof of relationship in either case.
You can alternatively try sponsoring her as a conjugal partner where you have to demonstrate relationship but not marriage.