Just build a 2x4 or 2x6 pen with a dino gate on one side, placing stone pillars on the outer edges all along it. Make it near the green obelisk! Then fly the griffin into it thru the door with a pteranodon, fly out thru other end thru the pillars, land your mount, run to the other side, close the door, boom. Trapped griffin. Then tame with roughly 150 tranq arrows. Use mutton or prime meat as they are easiest to come by.
Make a box with pillars and behemoths and leave gaps just big enough for u to go out then anger him and lead him in and run out through pillar and spam shock it then use prime raw or cooked and just feed bio toxin and make sure it’s under lvl 20
It will take hours if you feed it normal meat so feed it kibble or lamb chops
So first nock it out put Dino gates all around it (4 gates) and the Dino frame then you will have to stack pillars stone or metal to place a roof on the top.
Last time I did this then left the griffin by itself when I came back it was still in the pet but it was awakened.
You can feed any creature bio toxin and no heath will be lost but more torpor will be there than a narcotic.