Does the prayer of a worshipper become invalidated if he gestures with his hand to move away a woman who prayed next to him due to overcrowding, given that he has a reason for doing so, such as an allergy or pain?

Prayer is not invalidated by a slight gesture for a need. Evidence for this includes several hadiths, among them the Prophet's (peace be upon him) pointing with his finger when Suhayb greeted him while he was praying, his opening the door for Aisha while he was praying, and his carrying Umamah while he was praying.