Help with calculating frequency offset
So, the TCXO is driving a direct downconverter (quadrature demodulator). Since the carrier is at 1176.45 MHz and the TCXO has a 10 MHz output, the TCXO output is most certainly being used as the reference for a frequency synthesizer such as a phase-locked loop (PLL) - otherwise, the problem would not make sense.
Presumably, the PLL is configured to output the exact carrier frequency (1176.45 MHz). Since the PLL is referenced to the 10 MHz TCXO, the frequency error of 10 ppm propagates through. Thus, our local oscillator frequency range is:
$$1176.45\textrm{ MHz}\pm 10\textrm{ ppm}=1176.45\textrm{ MHz}\pm(10\times 10^{-6}\times 1176.45\textrm{ MHz})$$
$$ = 1176.45\textrm{ MHz}\pm 11.8\textrm{ kHz}$$
Since our local oscillator can be off by up to 11.8 kHz, this means that the carrier will appear shifted in the quadrature baseband by up to 11.8 kHz in either direction. Assuming that the 1176.45 MHz signal is not already offset due to frequency error from the transmitter or Doppler shift, we would expect to see the received carrier offset by up to 11.8 kHz (positive or negative) in baseband.