Does down-mixing a signal before taking the fourier transform aid in signal detection?
If you are doing are doing this in both the analog and digital domain, then there may be significant differences due to trade-offs between the noise sources in the analog stage(s) in the filters and the mixer oscillator frequency, and the corresponding sampling jitter and quantization noise in converting to digital.

Even strictly in the digital domain, the bandpass and lowpass filter responses may have different effects on how potential stop-band noise might affect your detection accuracy, compared to how windowing (rectangular or otherwise) affects the continuous frequency response of each FFT bin.