For a good, cheap DSP device with FFT libraries available, have a look at the DSPIC33EP512GP502-I/SP
Well under $10, 70MHz, available in 28 pin DIP, 512K flash and 48K RAM with I2S interface among many other functions.
https://www.microchip.com/wwwproducts/en/dsPIC33EP512GP502
Standard libraries are available with the various free Microchip development tools
For the older libraries to work on the '33 rather than '30 series you may have to change the build model to "Large memory".
The limiting factor is that it's an integer DSP device; it depends if that is sufficient for what you need.
There are some references to using the FFT libs on the DSPIC33 here:
https://www.microchip.com/forums/tm.aspx?m=126331
The 829 DSD files are no longer on the microchip site but I have a copy, attached.
That overall project is just a demo but it uses the FFT library, called from the "frequency.c" file.
(I had to leave out the audio samples, they make it too big to upload - I can put them somewhere accessible if needed).