How can I sound proof bathroom walls?
I wouldn't tear the bathroom wall apart... I'd tear the bedroom wall apart.  What you are experiencing is the noise traveling through mechanical connections to the wall (screwed to studs) and the noise from the water passing through thin copper pipe (probably the Type M pipe rather than the thicker Type L).  

The only effective way I know of is to insulate that wall with Roxul Safe and Sound.  It's an insulation designed for noise.... and noise only.  It doesn't burn.  To go the added mile, use resilient channels before hanging the drywall.  The demo is probably a DIY, the installation of the Roxul is DIY, and the drywall.... call someone.  For a few hundred bucks the repair will disappear.

Edit:
Blown in insulation will not work... blown in is for warmth, not sound. It's the sound transfer you're trying to conquer. We once built a house and insulated the walls with thermal insulation, thinking just like you are. It didn't work. We had the type L pipe, all the plumbing was fastened with cushioned fasteners. We thought we had it knocked. Turned out, it was a waste of money.

If you truly want to fix it, just take down the drywall and do it the right way with the Roxul. Going into a wet wall from the bathroom side is a suicide mission. Building an offset wall will create a door opening that's 10"... with a door that has a 5" jamb. Then there's the carpet... or other flooring surface. It sounds like a difficult job, but it's probably only one weekend.

Otherwise, just buy your son some foam ear plugs at Costco and call it a day.