I am an embedded software engineer with 6 years experience and I am familiar with STM32 Cube and STM32F4 Discovery. Before I knew STM32 Cube, I successfully ported FreeRTOS to GNU Arm Eclipse environment. It needs some tweaking before the FreeRTOS blinky LED running.
I know It takes some effort just to set up GNU Arm Eclipse with OpenOCD, and once you're successful, you essentially have the same funtionality with commercially packaged IDE such as Keil.
I've been involved in laser camera controller, also motor controller and smart pipette controller. I believe I can do this project. Outputting FFT to LCD is easy. It will take about 1 week, assuming I only work 10 hours per week. I take this as a side job, so I can't put more hours.
However, your documentation indicates that you might need other function than FFT. Am I right?