This 125kHz sometime goes through AM peak detection before reaching the op-amps. Possibly, coolrfid's creating a bandpass filter for the 12kHz and 15kHz FSK frequencies.
5V or even 400mV will be enough if your ± supply voltage is in the recommended ±5V to ±15V range
Since you're using a quad op-amp , I'll just wager you're using the TL084 to make Sallen Key filters and inverting amps. A lot of Sallen Key filters I find have unity gain. So if you're putting 400mV in, you should be getting almost 400mV out...unless it's outside your passband.
...That's just based on my guesses. First thing, post a diagram so we can see your amps/filters. Then tell us what frequency this 400mV is.
Also keep in mind the TL084 can't produce an output if the load impedance is under 100Ω.