Carrier frequency offset estimation for QPSK and asymmetrical spectrum
A likely source of the amplitude imbalance given it is on the actual signal is filtering distortion along the signal chain such that the gain is not flat across the bandwidth of the spectrum. This would not be uncommon with analog filtering and can be compensated for with equalization (if needed).

For an alternate approach to carrier tracking applicable to QPSK, consider estimating the carrier frequency offset using the baseband samples after timing recovery (timing recovery with a Gardner TED converges even under relatively large carrier offset frequencies). With that, the correct sample locations for each symbol (just prior to decision) are used with a cross product frequency discriminator to estimate carrier offset as follows:

$$Err = |I_2| |Q_1|-|I_1| |Q_2|$$ 

Where the absolute values convert all samples to the first quadrant (so this will work for offset frequencies such that the phase doesn't rotate more than $pi/4$ radians between samples, or up to 1/8th the symbol rate. 

$Err$ is proportional to frequency offset as the result above is the imaginary portion of the complex conjugate product of the two samples, which is proportional to the phase between the samples. A change in phase versus a change in time (the symbol duration) is proportional to frequency, since $f=d\phi/dt$.

Thus the cross product frequency discriminator is the cross product phase detector for the phase between two samples at two different times.