Or is there maybe a better way that is more user friendly?
Plus the strength of this approach is directly proportional to the level of reward. Without knowing what a person 'wins' if they successfully bypass your check, it is hard to know if the check itself is strong enough.
It ultimately isn't that hard to make a fake account and add three other random/fake accounts. Takes about ten minutes. Then wait a month. The question is, is the reward worth the hassle?
I don't have a better/alternative way of doing this. Online age verification is an on-going problem, and gender verification is a pandora's box (in particular related to LGBT issues).
Since gender is relevant, I assume you're making either a dating app, or an app that feeds into some sort of beureacracy. If the former, then people have an incentive to enter their real gender if they want to get any use out of the app. Linking to Facebook is a bad idea since people are usually kinda wary of linking their dating profile to other social media. If the latter, then facebook is not going to satisfy your stakeholder. You should consider asking for some form of ID or credit card.