Measuring THD using consumer sound card
The main prerequisite for measuring anything meaningful is that the target parameter (THD) is much better in your measurement system than in the system under test. The higher the difference, the more accurate your result will be. If your measurement system is orders of magnitude better, that the system under test, the measurement error can be neglected. If it is not, your result will have a considerable unsecurity. "Eliminating" the error introduced by your measurement system is not possible, because, like you said, it's inherently non-linear. 

For example: if the THD of your sound card, is just 10dB better than the THD of the system under test, the uncertanity will be at approximately 30%. With 20dB it will be 10% and with 30dB it will be 3% and so on.

Regarding your edit: No, this would not work well. As we said, THD is non-linear, not only in the sense, that new frequencies are generated, but also regarding the phase of those new components. E.g., if both systems have the same THD individually, it would not just add up (i.e. be 6dB higher), but could be anywhere from +6dB down to -inf. All information, that can be garnered from comparing the overall THD with the measurement THD comes with uncertainity regarding the individual values. That is a direct consequence of the non-linearity.