Or should I use methods based on what my GM says...
I believe this is the best answer.
Would I allow it? Sure! Griffons like horse flesh IIRC. A young developing creature probably couldn't control itself. I hope the party likes walking. A trained falcon might be easier to manage.
As hits and Exhaustion change with level, I'd alter OB too. Maybe it could be the Stamina Bonus; just use it for OB/lvl modifier. That might just be a House Rule.
The C&M book states that "if griffons are
intensely trained in their youth, then they will be a fierce, loyal steed, obedient to none save its rider".
As others have stated, a Training Package, Griffon Trainer, might be required. Don't be surprised if this requires around 2 years or more!
A
Tusker Goblin TP for riders of Battle Boars requires 52 months.
Boar Saddler -(similar to above but for Hobgoblins) takes 104 months.
Brelisker- The Hobgob version of lizard rider needs 94 months.
Also, I could see your GM saying that you might have to retire the PC while you intensely train the young cub.
If a female can only have 1/year, it might be reasonable to assume they need total care for the first year. Possibly additional care after that.
Or just ride a horse and pick a trained falcon.
Let us know how it goes. I'm always eager to hear of other people's games!