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.
Knocking out a high level lymantria within a single bola is impossible with a primitive crossbow. Instead, quickslot two pillars, two thatch ceilings, and five thatch walls. Bola the moth over flat land, then drop a pillar on one side. Ceilings next over it, then the other pillar on the other side. Hang walls from the ceilings. When it comes out of the bola, it's trapped. Tranq slowly so it won't die, and leave the trap up as a tame blind during taming. If you fail to trap it just try again; it won't flee far after a bola attack if you did no other attacks.