Are you allowed or permitted to have a black belt at an ITF affiliated school, and also train for a black belt with a WTF affiliated school? With the differences l have read on this site between ITF and WTF l can see the advantage of having both.
They are separate federations, and neither federation has a rule against training in a different style, even if they the same basic art (tae kwon do).
It would be a courtesy to the instructor (And also possibly enhance the training and approach), that you let them know that your son is also training in another style. Be prepared, however, in that some instructors will request that your son choose one style or the other.
The only thing that I would caution, however, is that your son is still a colored belt, and as such, learning the basics. There is a possibility that minor changes in techniques from one to the other may muddle his training and leave him with a technique that is somewhat in the middle of each style.
As to schools, the only rule against it would come from the individual school. But as to events, if it's WTF-sanctioned, and you've gone through a conversion process, it's perfectly allowed.
It's not uncommon for ITF students to compete in WTF events, and they occasionally train at ITF-friendly WTF schools. In addition, the USA-Taekwondo has opened membership to both ITF and ATA students, and until those students normalize themselves with WTF rules and practices, you'll see those students applying for membership at WTF schools as well.
As to Kukkiwon, the actual style that most WTF competitors hail from, they will recognize your dan and extend you basic courtesies. However, you would not be permitted to officially teach Kukkiwon curriculum or recommend for grading any student for the Kukkiwon dan process, nor would you be permitted to be a coach. To do any of these, you'd have to go through a formal conversion process, which up until recently was a fairly expensive process, but now is trivial.
By the way, I concurrently train in two schools - one Kukkiwon (WTF) and one ITF. Both are aware of the other, and even extend me the courtesy to allow me to cross-teach poomsae to the other. Although they can't technically stop me, getting the instructors' blessings goes a long way, since the Taekwondo community is small, and, occasionally, I need resources from one school or the other. and if they didn't approve, they could also stall my own instruction and dan grading process.