Bird species classification from field recordings remains challenging due to overlapping vocalizations and incomplete species labels. We study source separation as a preprocessing for bird species classification to improve multi-species detection. Specifically, we employ an ensemble of two separators, FTRNN and TF-Locoformer, both trained with mixture invariant training (MixIT). To address the false positive gain caused by separation errors in separated outputs, we propose mixture-constrained max pooling (MCM), which clips the predicted probability from each separated channel based on the corresponding species probability in the original mixture. The classifier is applied to each separated output and the original mixture independently, and MCM aggregates the predictions into a final per-species probability. Experiments on two real-world datasets show that the ensemble outperforms individual separators and MCM outperforms standard max pooling across multiple metrics, and reveal that separation leads to both true positive gain for present species and false positive gain for absent species.