I think you could use a USB-UART (sometimes called USB-TTL) adapter module and co-opt one of the serial control output pins such as DTR or RTS. Make sure the module you pick has access to at least one of the control outputs. This could be extremely cheap (a dollar or two from China or maybe $15-20 from Adafruit etc.) There are various chips such as CH340G, CH340E, PL2303, FT232RL etc., most or all of which have the control outputs on the chip but often not brought out to a convenient connector on modules since many applications need only TxD and RxD and power/ground.
That switch chip draws very little so you should also be able to draw the 3.3V from the USB bus via the adapter.
Here is some Python info regarding direct control of the handshaking pins.
As usual with direct connections to your computer, make sure no harmful voltages are connected to the circuitry, and consider the supply will be grounded through the USB in many cases.