Designing an analog circuit that Fourier Transforms or Laplace Transforms an input signal

I think you are going about it the wrong way. Think about what a Fourier transform does: It takes a time-domain signal and converts it to a frequency-domain signal. So what you want to end up with is a set of frequencies with individual amplitudes. Phase can be done also, but it's more tricky and rarely needed in my experience so I'll skip it for now.

Okay, so we want to take an analog signal on one wire and measure its amplitude in each of X frequency bands. Now it's easy. Just distribute the signal to X bandpass filters and measure the amplitude coming out of each. Like this:

^{simulate this circuit – Schematic created using CircuitLab}

There's even a full-documented project for this here: https://sound-au.com/project136.htm