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.

An easy was to tame a Lymantria us to use a bola on it then shot it in the face with tranq arrows or darts.

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.

Wait until night. Then build a light source it'll be unusually attracted to it.

Use 1 foundation, 4 walls, and a ceiling to trap it. Place foundation in front of it, bola it on foundation and build away