Ignoring the decision itself as it sounds like a definite YHTBT moment, I think I'd have taken the same approach to the caution that you did. Unfortunately at that age you're very likely to get an emotional reaction (e.g. crying).Now we've established it was NOT a dogso any other advice apart from not to caution him.
One approach that has worked for me in the past is to say something along the lines of "don't worry you're not going to be sent off, but let's go and have a chat with your coach". You can then walk over to the sideline with the player and explain the decision and the sanction, which has the dual benefit of the player hopefully reacting more calmly and, if not, ensures that everything you have said is in the presence of a responsible adult. Cautions are rare at that age and it isn't likely to be a regular ocurrence, so it shouldn't be too cumbersome.