I’ve gotten away with a 2x2 foundation trap 3 walls high and one Dino gate.
You can either :
1. run through the trap using door ways and run around to the other side where the door is once the bird is in,
2. or stand inside and when the argy is inside close the door then run out through the regular doorways. To do the second strategy you’ll need decent health (I do it with 350) and some armor depending on the level of the argy.
However since the trap is so small be ready for it to escape before the door fully closes.
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.