Instructions
Equip the syringe at the beginning of the fight with Screaming Mantis. Use the syringe on yourself to become immune to her ability to control bodies. Two dolls hover in the air near Screaming Mantis. She has a slashing attack that involves her teleporting to your location, so whenever she yells "SNAKE!" move out of that area to avoid her impending attack.
Equip your tranquilizer gun (called an "Mk2") and shoot at the Psycho Mantis doll hovering by her right side. She will control troops to attack you, so try to avoid them or tranquilize them to get them out of the way. She will then control Meryl, and you will lose the battle if Meryl is killed. Equip the syringe again and put Meryl to sleep. You can also put Meryl to sleep by shooting her from a distance with the tranquilizer gun.
Continue shooting at the Psycho Mantis doll as often as possible. Screaming Mantis will eventually take control of Meryl again. This time she will force Meryl to hold a gun to her own head. Run over to Meryl and use the syringe or tranquilizer gun to knock her out before she kills herself. Shoot the Psycho Mantis doll until it drops from her side.
Pick up the Psycho Mantis doll. You can repeat this for the Sorrow Doll on her left side, but it isn't necessary to win the fight. Equip the Psycho Mantis doll and use it on Screaming Mantis. Hold "L1" down and move the joystick back and forth to shake Screaming Mantis and cause her damage. She will eventually drop down to the ground. Use your tranquilizer gun on her to end the fight without killing her.