I'm not sure where you get 8.3784 from, 2* 4.1792 = 8.3584. Unless the wheel is pretty good, the chain will not mesh smoothly which could result in it coming off all the time, or snapping. Snapped roller chain whirling round next to grandson isn't an attractive prospect. Obviously you don't need to worry about the 2/10s of a thou, but 'ish' probably isn't good enough. I very much doubt that 3/8 pitch chain has 3/8" diameter rollers, depending on what the exact spec is it could be 5mm, 0.2", 6mm, 0.25"
You can get 71t sprocket and a length of chain for <£20 from this outfit
http://www.ebay.co.uk/itm/Kart-Gorilla-O-Ring-Chain-Sprocket-Offer-Best-Price-Rotax-TKM-/130962346758?var=&hash=item64289b7af2
I've no idea what pitch it is, but got to be worth a phone call.