There's no way an under 13's rider will be quicker than the next kid just because he has a downhill bike. Wait a few years for him to get more nerve then jump on a downhill bike. I'd totally steer towards a longer travel trail bike purely for versatility.
In a nutshell, modern enduro bikes are as good at downhill as downhill bikes were a few years ago. Adding the fact that you can pedal the thing back up is a bonus that he'll enjoy. Don't get me wrong, a downhill bike is a faster bike on a downhill trail but none of the trails raced in Australia need a downhill bike bar a few, especially none of the trails in the NSW series. You'll likely find the older of the junior guys are winning downhill races on enduro bikes because the trails used are better suited to it. Also to note is a downhill bike can feel like a waste of money under a light human as the air pressures and / or spring used can be too heavy at a minimum making the bike completely stiff. I've seen it a few times where a kid is using one third of the travel and getting thrown around, it's a waste of time and money until he takes his adult shape to a point.
A bit more travel than the 120mm he's using will help but throwing the most flash thing available at a kid does not mean instant results and can be completely demoralizing. He's young, he's got forever. Get a good trail bike.