The widespread availability of code-mixed data in digital spaces can provide valuable insights into low-resource languages like Bengali, which have limited annotated corpora. Sentiment analysis, a pivotal text classification task, has been explored across multiple languages, yet code-mixed Bengali remains underrepresented with no large-scale, diverse benchmark. Code-mixed text is particularly challenging as it requires the understanding of multiple languages and their interaction in the same text. We address this limitation by introducing BnSentMix, a sentiment analysis dataset on code-mixed Bengali comprising 20,000 samples with 4 sentiment labels, sourced from Facebook, YouTube, and e-commerce sites. By aggregating multiple sources, we ensure linguistic diversity reflecting realistic code-mixed scenarios. We implement a novel automated text filtering pipeline using fine-tuned language models to detect code-mixed samples and expand code-mixed text corpora. We further propose baselines using machine learning, neural networks, and transformer-based language models. The availability of a diverse dataset is a critical step towards democratizing NLP and ultimately contributing to a better understanding of code-mixed languages.