Destination NAT onto the Same Network in AWS using IPTables
The issue was the FORWARD chain was not allowing return traffic so the following rule was needed:
sudo iptables -A FORWARD -m state --state RELATED,ESTABLISHED -j ACCEPT
The full solution looks like:
sudo iptables -A FORWARD -p udp --dport 53 -d 192.168.1.151 -j ACCEPT
sudo iptables -A FORWARD -m state --state RELATED,ESTABLISHED -j ACCEPT
sudo iptables -t nat -A PREROUTING -p udp --dport 53 -j DNAT --to-destination 192.168.1.151
sudo iptables -t nat -A POSTROUTING -p udp --dport 53 -j SNAT --to 192.168.1.213