If you have access to a snow owl, you can tame these with any crossbow! Just shoot it until it looks bloody, and heal it with the snow owl’s ability. Do it so that it heals quite a bit, but not so long that the torpor drains back empty. Then hop off the owl and continue shooting it with tranq arrows. I knocked a 140 with a 279 crossbow and using this method it had over half hp remaining once it finally went down. Help this get to the top!
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.