Can someone with more experience with these stuff recommend me the best approach to the problem?
I would get an MCU which has enough pins and a built-in USB module. As I recall, FT232H is $4 and requires an oscillator ($1). You can buy a good MCU for this.
Or, if you can't part with your existing design, get PIC16F1454 ($1, no oscillator required) and program it as an USB to SPI bridge.
Of course, if money is not important, FT232H will work.
FT232H also has GPIO pins. It doesn't have programmable brains, but you can manipulate the pins directly from PC. Thus if your design requires PC connected at all times, it might be possible to get by with FT232H alone, without any MCU.