Help wiring a Raspberry Pi and odd DB9 connection
The DB9 breakout board that you specified is also a level-shifter. Supply it with 3.3V from the RPi. Then you can connect it directly to the RPi GPIO pins, no resistor is necessary.

The hardest part is getting the signal direction correct (3.3V and RS-232). TX &amp; RX are only meaningful when the point of view is known.

The RS-232 DB9 cable may need to be a crossover cable, or you may need a null modem adapter.

Your level-shifter only has RX/TX, no handshake. You could substitute the signals and make a custom RS-232 cable. 

The custom PC software is apparently using Data-Set-Ready to trigger.

You have an RS232 level shifter, that may or may not have the handshake signals wired, I can't tell. But we can use the TX signal instead.

You will need to make a custom cable.

I can't be sure that I got the directions correct, it would help if I had full documentation for the level-shifter. Even with all the documentation, it is very easy to get confused, I often had to scope signals to get COM stuff working.