A simple fix(more of a workaround) for this that I can think of is potentially using routing rules for the game servers - any connections to those servers will be routed specifically through the 4G gateway. By standard configuration, the other programs should use wired anyway. If not, I believe there are priority rules you can look at.
I should also point out that this is an EXTREMELY hacky way of doing it that may not even work - however, i haven't managed to force a process onto a certain NIC in the past, so I have no native ideas that work apart from maybe this one.
I can also recommend you actually look at your router settings and discover why your ping is so large for your wired connection. This essentially indicates it takes long for the ping to go and return to your computer from the target server. Its possible you have a very low bandwidth and its being saturated - try ping tests when no one else is using the internet connection, and if it gets better, then it means your allocated bandwidth by your ISP is too low - at least you can pester them to increase it :)
If the ping doesn't change, it just means your device is too far from the server which introduces the constant latency. (this is all assuming you are also connected directly to your router - things like powerline adapters and ethernet repeaters can all introduce some sort of latency)