Try shutting off "Ask to Buy" and see if the App Store goes away. The "Ask to Buy" and the "Installing Apps" settings are somewhat conflicting because when "Installing Apps" is set to Don't Allow, then the App Store should not be available on the device, but with "Ask to Buy" turned on, the App Store must be there for the child to request the purchase of a new app.
After an app has been allowed to be downloaded it will appear in the App Store to be able to be redownloaded. The only solution I see is to remove the App Store, which might require shutting off "Ask to Buy". There does not seem to be an easy way to block a specific app after it has been downloaded as long as the App Store is available on the device.