I've posted most of this elsewhere, so my two cents:
I started judo and TKD at the same time, when I was 11. There was no overlap or interference at all, and both teachers encouraged my participation in the other.
Three years later, still going at it, I was exposed to kyokushin karate in boarding school, and it was all that was available to me, besides boxing, wrestling and fencing. So I did kyokushin at boarding school, and TKD and Judo when I was home-luckily, this was long ago enough that the TKD forms were variants of the same kata I did in kyokushin-so it took adjusting and remembering where I was as far as hand placement sometimes, but I did okay with both,
with the consent of both teachers-earning dan grades in both-until I had to decide to give one up. I also wrestled and boxed for a time, though I didn't stick with fencing......
Short answer-especially for a kid-dissimilar forms like a grappling form and a striking form aren't going to cause much interference with each other at all. Striking forms of something as similar as TKD and karate can be might be confusing-.
There is a lady in Albuquerque who holds ranks similar to mine, and
her program consists of Okinawan karate and TKD, and her kids' program works well, but I think she's rather anomalous in that regard.....