Lure to taming box and tranq til flyes other direction then drop dead fish body and wait til health ring appears then knock it while health ring is over head, hunger goes all way down and makes it a 30 second tame.
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.