It is certainly doable to go self-contained. I would like to strongly encourage you to do several multi-day trips before hand, perhaps a week-long trip over spring break. I was 14 on my first "big" tour (1,000 miles) but had been on a couple of supported cross-state rides before then. You don't mention if your son is a strong rider or an average rider. The amount of gear you need is not that different no matter the size of the person. I would assume your son would have a little more difficulty lugging 30 pounds of gear around compared to you. Of course, a self-supported tour is easier to pedal but you loose a ton of freedom.
I would also strongly suggest you adhere to the rule "Each item must be at least double duty." This means, you do not take an item if it can not be realistically used for two things, i.e. your off bike shoes can be Crocs, i.e. you can walk around in them, shower in them, and use them as a "pillow". Another example is that your rain jacket is your cool weather jacket. Obviously, some critical items like a stove are rarely multi purpose, but try to keep the rule. Otherwise, you end up shipping a ton of stuff home a couple of weeks into the ride.
Since you have all summer, IF both are strong riders, I would encourage you do to the TransAm. It is easily doable during summer break. I did it the summer after my junior HS year and loved it. It is very educational, the scenery is great and changes often, and it has been around for over 35 years so the locals are used to cyclists.
Whatever trip you do, I wish you an enjoyable time!