What you are looking for is called "NAT Hairpin" or "NAT reflection" or "NAT loopback"
It may not be supported by the customer gateway, and the exact way to configure it depends of course of the box used as gateway.
Quoted from this article in Wikipedia
NAT loopback, also known as NAT hairpinning or NAT reflection, is a
feature in many consumer routers which permits the access of a
service via the public IP address from inside the local network. This
eliminates the need for using separate domain name resolution for
hosts inside the network than for the public network for a website,
The following describes an example network:
Public address: 203.0.113.1. This is the address of the WAN interface on the router.
Internal address of router: 192.168.1.1
Address of the server: 192.168.1.2
Address of a local computer: 192.168.1.100
If a packet is sent to the public address by a computer at
192.168.1.100, the packet would normally be routed to the default gateway (the router), unless an explicit route is set in the
computer's routing tables. A router with the NAT loopback feature
detects that 203.0.113.1 is the address of its WAN interface, and
treats the packet as if coming from that interface. It determines the
destination for that packet, based on DNAT (port forwarding) rules for
the destination. If the data were sent to port 80 and a DNAT rule
exists for port 80 directed to 192.168.1.2, then the host at that
address receives the packet.
If no applicable DNAT rule is available, the router drops the packet.
An ICMP Destination Unreachable reply may be sent. If any DNAT rules
were present, address translation is still in effect; the router still
rewrites the source IP address in the packet. The local computer
(192.168.1.100) sends the packet as coming from 192.168.1.100, but the
server (192.168.1.2) receives it as coming from 203.0.113.1. When the
server replies, the process is identical as for an external sender.
Thus, two-way communication is possible between hosts inside the LAN
network via the public IP address.