I've had a bit of a look around specifically at serial ports and it seems it might be quite difficult to get an old-school RS232 PC serial COM port to run at a non-standard baud rate, unless I'm missing something. (I realise that for you, 'difficult' is a novel concept).
In other situations (PIC microprocessors would be a good example) you have a freely programmable UART clock which is often taken advantage of in order to generate a non-standard baudrate of 31250 (For MIDI), and on the Raspberry Pi it is (was?) possible to alter the clock feeding the UART divider chain so that setting a near baud rate such as 38400 in, ie, terminal software, results in an actual baud rate of 31250.
Do standard PC serial ports lend themselves to being wrangled onto non standard baud rates without having to go straight to the hardware registers? I have to admit I have never looked into it.
(The newer USB serial comms cables / leads might be a lot more flexible in this respect).
Edit: Just came across this, which seems to suggest that FTDI based serial USB devices allow 'soft' reprogramming of one standard baud rate with a non standard one, so that when software selects the replaced baud rate it will actually select the modified baud rate.
https://www.ftdichip.com/Support/Doc..._BaudRates.pdf