Dual NIC on Busybox: How to set up a global default gateway bound to just one of the NICs?

I have a network appliance based on Busybox (on a Zynq device), and it has two NICs. One network has access to the internet, while the other is private. Both have DHCP, so both are reporting gateways, even though the private network doesn't have Internet access. For some reason, traffic for things like ntpd, rdate, and ping is being routed to the private network. We need to get NTP working, but the Busybox ntpd doesn't have a -I option. We've tried deleting the gateway (route del default eth1, IIRC) from the NIC for the private network, but that results in making the device completely inaccessible.

At a minimum, I need to figure out how to just get NTP to work. More generally, it would be nice (but not critical at the moment) to figure out how to set up a default route for all outgoing traffic to go to the public network if it doesn't match the IP range of the private one. Any help will be most appreciated.