How to configure traffic from a specific IP hardcoded to an IP to forward to another IP:PORT using iptables?

Your NAT rule seems ok, but have you also added an ACCEPT rule for this traffic?

Otherwise, NAT will kick in finely, but then the resulting packet will silently be dropped...

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Update:

You need this rule in the FORWARD chain (which is where packets going thorugh the firewall but not directly related to it are processed).

FORWARD rules are applied after PREROUTING rules (like NAT), so when this rule is applied the packets will appear like they're coming from the original IP but directed to the modified new destination IP. The rule should thus be somewhat like this:

/sbin/iptables -A FORWARD -s $CUSTIP -d $NEW_SERVER_IP -j ACCEPT