You can tell with pigeons, even without a genetic test, and often before they are mature, if you can hold them in person. Utility breeds can be tougher, but still possible.
How is your pigeon sticking her beak in your mouth an indicator? Both male and female pigeons do this.
@Toxicity 321 , can you post a video? That could help. Fighting could mean two males, or an un-bonded pair. Can you separate them a week, so they can't hear and see each other? After the week, put them back together and see if they are excited and begin courting, or excited and begin fighting. It's likely they were separated from their mates when sold, so they may be waiting to see if they can get back with their mates, before repairing.