For an easy tame, wait for the Lymantria to land and then place a tent directly where he/she is at. Its stuck until the tents durability dies out, this gives you enough to build a little house around it for a safe, easy tame.
Don't attack it unless u have a plan a good one is to wait for it to land, throw a normal bola at it, then shoot tranq arrows at its head until it passes out, then feed it mejoberries or any type of berries, you may want to build a house around it to protect you and the lymantria from storms
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!