Roughly comparing between a lion and a leopard, I would make the following adjustments to the base manticore to get a CR 3 manticore cub:
Reduce HD to 3, adjust BAB, base saves and skill points accordingly.
Remove Weapon Focus (spikes) feat, adjust attack bonus accordingly.
Drop size from Large to Medium, adjust AC, attack bonus, grapple bonus, base damage for bite, claw and tail spikes, and space/reach accordingly.
Reduce Strength by 4 points, adjust attack and damage bonuses accordingly.
Reduce natural armor bonus by 2, adjust AC accordingly.
Reduce the number of spikes it can release in a volley to 3, reduce the total number of spikes it can release in a 24-hour period to 12.
To be sure, you could run it through a simulated combat with a level 3 party and see how it fares.