Well, you're not getting off this easy.
First of all, what you call "public" is not at all public, just another network. Label them accordingly. Am I tight that the devices on the right are configured with their addresses as 192.168.1.x/24 and have their default gateway set to 192.168.1.1? "And" as in are both of these true?
Am I also correct that on the left (public) side the devices are set to 172.22.202.x/24 addresses and have 172.22.202.1 as their gateway? Presumably there is a router (not shown) that is also on this subnet and deals with any further connectivity?
Is it also true that there will be another device (also not shown) on this 172.22.202.x subnet that will provide you with vpn access? Or is this the same device as mentioned above (the not shown router)?
If all this is true then you're looking at a fairly easy job.
My only worry is this: Am I correct that the WLAN 1120 device is the FL WLAN 1120 module from Pheonix Contact? I don't really have experience with the specific device, but it seems to me that only one side is wired ethernet and the other is WiFi. So... the next question: is the connection to the right (to the hEx S and the switch) the wired one? Is then the other connection (up, towards the cloud) WiFi?
If it is indeed WiFi as I suspect, you may run into some difficulty. The standards compliant (802.11) version of WiFi doesn't support multiple devices behind a single client. There are various workarounds implemented that work... in various ways. The most reliable one is where the access point and the wifi client are both made by the same manufacturer and are documented to work in these situations.
Is it possible for you to temporarily link the hEx to the network using a wired connection (potentially via switches) temporarily? It's much easier to troubleshoot these sorts of things one difficulty at a time.
Update your device to some sensible v7 version. It is fully supported. Although the 6.x versions support everything you need, it's really unnecessary to dig yourself into a legacy hole from the start. (If you would have a fleet of devices already configured and working well, I would suggest the opposite: there's nothing wrong with v6. You'll just find way more documentation for v7, and the newer devices only support that.)
EDIT: Forgot to mention: there's nothing wrong with having the older hEx. It's a thoroughly tested and well respected model.