If you beat the youngster with the slowpoke can you still catch a mew?

Yes, use the Ditto Mew Trick

Catch a Pokemon with a Special Stat of 21. Fly away from the gambler on Route 8 and when you fly away a ! should appear above his head. Then fly to any trainer and battle and beat him. Then fly away to anywhere that has a Ditto in the Wild and when you encounter one let it transform into the Pokemon that you have with a Special Stat of 21. Beat the Ditto, DO NOT FACE ANY POKEMON AFTER THIS. Then return to Route 8 and you should face a Wild Mew.