Invest in the best most expensive shoe you can find. Doesnt have to fit. Then make sure the kid can do a v5 at the gym with bat hangs only by age 3 and you'll never worry about paying for that shoe ever again. Gotta throw it on YouTube tho.
My kids use sandals that have the rubber/sole come up over the toes. Something like a pair of kids keens sandals. Most kids routes have pretty big footholds. Its working so far with my 10 year old up to 5.10 in the gym and 5.8 outside. She doesn't climb that frequently so she would probably only get 1 or 2 uses out of a climbing specific shoe before she outgrew it.