A lymantria can fit through a single ceiling tile space but an argentavis can't. Build a trap with one ceiling missing in the corner. Use a lymantria to aggro an argentavis, fly into the trap and out the hole, leaving the argentavis in the trap. Easy argentavis tame.
Lymantrias typically are nonagressavive until attacked. Rather than a typical attack they use a large cloud of green acid that will temporarily slow your movement speed. The absolute best way of taming would be the bola method. To know our aim towards darts rather than the typical crossbow or stone and slingshot. Seeing as of these creatures have such low health. Happy taming!
Place a pillar over the lymantria, build a small shelter around it to keep it safe. Use a bola when it lands, shoot it with tranq arrows with a normal bow. Crossbow has higher chance to kill it. Feed it mejoberries.
I would highly recommend building some sort of trap when taming a high level. Otherwise, without a good tier weapon, even a bola won't prevent a high level from breaking free and flying off before you knocked it out! ~L