Catch them in an argentavis trap. Get a magnifying glass. Shoot them with narc arrows until it is just about to go unconscious (Use the magnifying glass to inspect the torpor) feed the argentavis a body. Imediately after it consumes the body shoot it with narc arrows to finish knocking it out (*if it does not have the green effect around it's head do not knock it out! If the effect is gone feed it another body) Purpose when the argentavis gets knocked out with the effect his hunger will already by very low. Super fast taming!
An argentavis can't fit through a ceiling with one tile missing, but a pteranadon or lymantria can. Build a trap with one corner ceiling missing, aggro the argentavis, fly in one side and out the missing ceiling tile, and land. Dismount, run around, and close the entry door. Tame safely at your leisure.