Compensating Saturation Voltage of a transistor
Increase the resistors R1, R2 by at least an order of magnitude and decrease the capacitor proportionally - try maybe a few hundred pF.

Also decrease the value of R3 to maybe 1-2K, and you should be able to get Vce(sat) down to tens of mV. 

You can also replace the transistor with a small MOSFET and the then you'll have zero offset and just Rds(on), which can be very low (even a 2N7000 will have only a few ohms resistance, and you can get parts with much lower on resistance).