I was thinking the same thing for my 5 year old. I/he runs his 2.25″ Small Block 8s REALLY soft and it does most terrain pretty well, great on rocks/roots etc… but as you say, pretty crap in mud. If it’s wet mud it’s OK but they’re f’ed in sticky mud. Spawn Cycles have a cool 20″ tyre, they used not to sell it in UK but now their site has a Europe option, the tyre is called the Brood Maxtion 20″. I think there’s a tyre called a Vee Crown Gem that comes in a bunch of different widths up to 2.8 – I think Commencal may spec it in 2.8 on their kids mountain bikes. Think Tredz sell that one.
I reckon the Brood one looks the best, but it’s 40 euros. Probably good on the front with the Small Block 8 out back I’d think so the whole thing doesn’t roll too slow.