For low frequency problems as your application, the terms injected and returned signal should be better replaced by voltage and current in my opinion. You inject a current and measure voltage or vice versa.
You can refer to the design of state-of-the-art LCR meters, how they achieve a precise measurement. I think, the most simple and straightforward method is synchronous quadrature demodulation. FFT is a more general approach, although I don't exactly see an advantage for this application.
For the possible system topology, should also take a look at the Analog Devices AD5933, a signal chip DSP impedance analyzer. It uses FFT.