Not sure how much help I can be with all this. I don’t have any experience setting up one of those devices. Where are you actually running into problems? Do you know? Are you getting connected to the local network? Are you getting a socket connection to the server? Are you getting any sort of response from the server?
Once you get to the point that you’re connected to the network you should be able to more or less copy what our Arduino library is doing in this function: https://github.com/exosite-garage/arduino_exosite_library/blob/master/Exosite.cpp#L58
And actually, if that TcpClient library is compatible with the Arduino TcpClient you should be able to use or library simply by passing in the client object to our library. (I’m not sure if it is because I can’t actually find anything called that.) But that still depends on you setting up the network connection before the library is able to use it. For assistance on that I’d suggest contacting the vendor.