I say, and with God's success: If he is praying a voluntary prayer, he can perform two additional rak'ahs if he wishes with a new takbir. If he is praying a mandatory prayer, it does not harm him to greet after the first two rak'ahs as long as he does not engage in anything that contradicts the prayer, such as speaking or eating or otherwise. He then stands for the third rak'ah and completes his prayer without a takbir and prostrates for forgetfulness at the end of the prayer. And God knows best.