Your root problem is likely "First, I tried to applly the samppling and filtering to obtain the modulating signal."
The first step to obtaining the modulationg 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!
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 good insight. Then start tweaking up the data rate!