Once trapped in a dino gate trap, use tranq arrows carefully untill the argy starts trying to run, then throw a dead dino such as a dodo or monkey and let it eat it, then light it up with tranqs till its knocked out. When the argy is KOed its hunger will have diminished to a almost starving level, go kill some dinos that drop prime meat, you have about 4 minutes to get the meat and feed the argy, this is the most efficient way and it also adds to your tame effectiveness leveling the argy even more.
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.