The DTR is definitely for Arduinos - I used a similar board and just cut the trace going to the "DTR" pin on the end and used a jumper wire to connect it to RTS from the computer instead. If that link you provided is the board you have, then RTS is available on one side of the board and it looks like the cut&jumper method would work for you. Some of those boards even come with jumpers on the back to let you select between DTR and RTS on the last pin.
One note when using FTDI chips with handshaking is that they do not stop sending characters the instant you deassert CTS. They can send up to 3 more characters. This is only mentioned in a support FAQ for the FTDI232 chips and it took me by surprise because I was expecting no more than 1 additional character after telling the PC to stop sending.
From
https://www.ftdichip.com/Support/FAQs.htm: If CTS# is logic 1 it is indicating the external device cannot accept more data. the FTxxx will stop transmitting within 0~3 characters, depending on what is in the buffer.
I ended up going with an interrupt driven receive with a circular buffer and signalling early if I needed the PC to stop sending.