Cannot communicate with serial device via an Arduino Nano serial communication
Most likely there are multiple points of failure here.
First of all, the RS232 to TTL converter TX pin is an input for the RS232 transmitter, so the TX input must be connected to TX output pin of the Arduino. Likewise, the converter RX pin is an output from the RS232 receiver so it must be connected to Arduino RX input pin.
The RS232 to TTL converter DE-9 connector is a female like any other DCE, so the pin 2 is RS232 TX output, and pin 3 is RS232 RX input, and it is meant to be directly connected with straight through cable to a DTE like a PC. So when a DCE is connected to another DCE like the reader, it means that a crossover null-modem cable is needed, to connect TXD of one device to RXD of another device. Simply an adapter will have direct connection and will not work.
Then the final issue is that the Arduino is running at a 16 MHz clock. Also, the clock does not come from a crystal, but a less precise ceramic resonator, which is only accurate to about 0.5% initially and can deviate with temperature to about 1%.
Now, I don't know the details how the softwareserial works, but depending on what other things the program does, it may have problems running at 115200 rate reliably. Even using the hardware UART will not help much, as 16 MHz clock just does not allow for 115200 baud rate below 2% of error in the rate, and adding the ceramic resonator error, it may just result in so large error that it does not work at 115200. Suggested maximum error rate is within 2%.