Differential ADC for Load Cell Measurement or In. Amp?
A 20mV output should not be too noisy so I reckon it's the layout issues you have and a proper PCB should cure that. I'd definitely be wanting to analyse the peaks inside the MCU as it gives you many more options.

Whether you use an ADC or instrumentation amplifier you should consider low-pass filtering the signal before it goes into an ADC input to improve the noise situation as much as you can. You'll also need to consider putting an anti-alias filter inline too. Both filters are easily combined.

If you do sample at a much higher rate than what you actually need you should also consider an averaging filter in software too.

What load-cell excitation voltage are you currently applying? Maybe this can be increased to get a bigger signal?