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.
I tamed the lymantria in early game valguero and it was actually quite easy. All you need to do is use a bola (I used a normal one) and then keep attacking it with a club and it should get knocked out in couple of hits. However, if your trying to tame a higher level one, bring multiple bolas. The saddle is quite cheap so it can be utilised early game.
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 150 can take multiple tranq arrows from a master craft cross bow last time I tried to get a high level I bola it so there I am with my master craft cross bow shooting it well it eventual got free so I go back for round two I do the same thing and the same thing happened it got free and flew away
The way I knocked my first one (don't tame when the raptor rockdracke guys are close) is with a bola and a club they eat berry but they cannot collect, There only Attack is a spore type attack, they slow down who, or whatever is in its radius...
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.
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