Yes, of course, that would be the lock-in scheme, with superb noise suppression characteristics. Unfortunately, offset voltage of OPamps would again come into play and so, you would need such a synchronous demodulator at each of the four outputs. Then, you would add an inverter and choose the true or inverted signal by the help of CD4066 for instance, and would do this with the same rhythm as the BC337 is toggling.

Kai