**** getting the lightest weight bike or branded suspension, you just need one with a back brake that does awesome skids! 😉
But.. if you want to get a bike that will add cmfort and confidence fit decent tyres, look towards BMX tyres rather than kids ones.. setting the bike up to fit them properly makes a difference too. this includes bar/lever angle, stem height and reach, setting up the levers to bite close enough and bending the brake springs to have a lighter lever feel.. other than the tyres all of this will cost you nothing!
oh.. and while I’m at it..
Any cheap sus fork will just add weight and pogo.
is not strictly true!
if you sort out the spring weight to sag appropriately for a child they only need a very slight amount of damping, this can be achieved in a number of ways. You’ll need to bodge a little but by using elastomers, oil etc you can get some kids forks not to pogo any worse than a RS fork from the 90s (no-one complained about it back then).
I fitted My daughters 20″ Hotrock’s fork with one spring (they come with 2) with an elastomer stack inside of the correct weight and it worked really well.. perfect spring weight and never rebounded too fast or uncontrolably (she’s lucky enough to have never felt fork top ;)). I could easily have gone further and added a crude damper in the other leg but she was so light it jut wasn’t needed.