Impressive that your winterbike, even 1×9 (XT shifter/fr mech+ 2 rings = 120g+140g+30g+40g = 330g=less than 3/4lb in weight saving) is less than 21lbs. I assume its carbon forked rigid?
The short answer is unless you move to using very high spec parts (xtr/x0)you have little or no chance of getting under 21lbs. and then you would need to go carbon rigid or high end/spec 80mm xc fork (SID/Magura Durin)and use light wheels.
The Islabikes have a light frame already so its component choice that will bring it down further on any comparable bike, and given frames on an adult bike account for less than 14% of bike weight on a 21lb bike its always the components that will make the difference.
Not much on a bmx to add weight!
So if you want a sub 9kg childs bike its going to cost you much the same as an adult bike.
Having looked myself there isn’t much out there that will fit with highend componentry. if he’s tall you could look at a more traditional trials/street trials bike 26″ or 24″ (Curtis T1 but preferably lighter) with a rear mech hanger as even with a 20T front the gearing is fine for kids; 20/11 gearing being similar to 32/16 SS.