Call me a retro grouch, but this is one of those areas that I think that bike would be better served by a set of mechanicals. Don't get me wrong, I'm all about hydros.
At 3, power is not needed. It could be argued that discs aren't needed, but that's besides the point, and well, discs are already on the bike.
With mechanicals, the levers can be adjusted in with the set screw ( I even placed a small plastic cap over the end of the set screw on my sons bike years back when the set screw wasn't long enough, worked perfect). Then the caliper can be adjusted to have the fixed pad as close as you can possibly get it to the rotor. Preloading the mechanical arm a smidge before clamping the cable (provides better feel than using the barrel adjuster) will give you the free throw that you want. Just need to do a little trial and error.
The issue with trying to over fill the system is that you're asking for trouble/leaks. And trying to advance the pistons for less pad clearance is a hit or miss endeavor. May feel good this time, may feel like crap the next. With mechanicals, it's not going to go out of adjustment...you're not trying to balance fluid in one part of the system. Plus, kids at that size dump the bike, and they dump it often. The hits to the end of the lever are hell on the seal around the piston rod.
If it were me, I'd pull those brakes and put on a set of mechanical levers/calipers.
I do agree that the Junit bars are a good idea.