LF-RFID demodulation
Your root problem is likely "First, I tried to apply the sampling and filtering to obtain the modulating signal."

Your first step to obtain the modulating signal should be envelope detection.

You should also be aware that your ratio of carrier frequency to data rate (125k/20k) is fairly low.  This is a key issue because the resonant circuits in RFID often have narrow bandwidth. (Due to their high Q-factor) 

So check that your resonant circuits (TX and RX) have sufficient bandwidth to pass your data signals without too much distortion! You can do that by modulating switch SW1 in your diagram, increasing from a low rate and then noticing where the received signal power is down to 1/2 (or amplitude down by 1/1.414)

If at all possible, start with a very low data rate, e.g. 1k bits/s. That will make your life a lot easier and quickly provide valuable insight. Then start tweaking up the data rate!