Straight-through RS232 cables are used to connect DTE (Data terminal equipment, like a PC) to DCE (data communications equipment, like a modem or other device like your FC). So a straight through connection is what you require.
Null-modem (or Cross-Over) cables are used to connect two similar (DTE to DTE, or DCE to DCE) types of equipment together; typically two PCs. The difference is that the receive and transmit wires have to be swapped around, as do the control wires like DTR, RTS etc.
Anyway, you need a straight through connection. That is what your USB-Serial converter gives you.
Your problem is likely that the USB-Serial convertor does not give the full RS232 voltage levels. Most such convertors do not. Instead, they rely on most equipment just using logic levels, which is often all they need.
Your FC though says it needs the full +/-12V signals, probably because it uses them to power the serial communications (it is possible to extract a little power from DTR/RTS if the proper levels are used). That is quite unusual I think for a commercial device, but the pdf does specify the 12V levels so it looks like it is the case.
Can you try an older PC with a real serial port? And a straight through cable.
Failing that, talk to BK and tell them that they need to join the modern world and use a USB interface!