I agree about the need for trig before taking calculus. Don't skip that.
I'd also add that high school mathematics courses often lack rigor, though they try to make up for that problem by piling on more homework. Seriously: it looks harder because the kids are always working. My eldest has done math classes in different schools, and the classes in two public schools (one middle, one high) were jokes. The problems in his geometry class were all easy-type problems and the high school pre-calc class was the same. There's also what I call the scattershot problem: the problems in the book aren't cohesive and don't build logically on one another. They just race off in different directions.
In between these two classes, he'd done a rigorous algebra 2 course with me at home (3 chapters of trig), and he had to put in a lot of effort to do well.
Now he's taking second-semester calculus at the local community college. He didn't have any real problems with Calc 1, but he's struggling more with Calc 2. Part of the problem is something that HowlerKarma has mentioned, which is that he's having trouble with the idea of struggling to learn. Those too-easy math classes didn't help, and by this January, Algebra 2 was a distant memory.
My point here is that an AP calculus course may not lack rigor (though it will probably have the scattershot problem), will certainly involve piles of work (including a lot of summer homework), and may present its own set of unexpected difficulties.
Have you considered alternatives? He could do an AoPS or other honestly-rigorous pre-calc class over the summer and then coast through the Honors class in school (which will also involve summer homework, but at least it'll be easy). He could take statistics. If you have time, you could teach him (or learn with him) offbeat but important in-depth topics in mathematics, like Cardano's solution to a depressed cubic (make sure you buy a block of clay), Greek/Egyptian methods, and/or complex numbers. Stuff like that fills in a lot of background and can make the subject come alive in a way that Larson's Umpteenth Edition of Pre-Calculus never will (ever).
Personally, I think it's best to go into calculus with a SOLID background of knowledge rather than approaching it with the idea that you can backfill any potholes as you go. That approach works for lower math, but IMO, not for calculus.
