The approach would work (summing the losses and backpropagate through the sum), but it’s probably not necessary, if you don’t want to finetune below the auxiliary classifier.
Assuming that all layers are frozen (do not require gradients) except the last linear layer, the auxiliary loss would just vanish as it’s not needed.
In the original paper the aux loss was used to “enhance” the gradients at this point. I.e. the loss from the model output would be used to calculate the gradients for all layers. Since the model is quite deep, the gradients tend to vanish in the first layers. Therefore an auxiliary was used to create another gradient signal and to sum it to the “output signal” to have valid gradients up to the first layer.
In your case, if the layers are frozen, the aux loss won’t do anything besides being calculated.
Correct me, if I misunderstood your use case.