How do I send data over TCP from a SIM5320A?
Turns out that the issue that I was having is that I was not using the commands AT+CGSOCKCONT=1,"IP","CMNET" and AT+CSOCKSETPN=1 before opening the TCP socket.

If you are trying to get the SIM5320 to open a transparent TCP connection, the commands you need to send are:

AT+CGSOCKCONT=1,"IP","CMNET"
AT+CSOCKSETPN=1
AT+CIPMODE=1
AT+NETOPEN
AT+CIPOPEN=0,"TCP","TargetIpOrDomain",TargetPort

When you are ready to end the session, you send +++ to the chip to enter command mode. To close the connection you use AT+CIPCLOSE=0 and AT+NETCLOSE to turn off the network connection mode.