Sentiment Analysis (SA) models harbor inherent social biases that can be harmful in real-world applications. These biases are identified by examining the output of SA models for sentences that only vary in the identity groups of the subjects.Constructing natural, linguistically rich, relevant, and diverse sets of sentences that provide sufficient coverage over the domain is expensive, especially when addressing a wide range of biases: it requires domain experts and/or crowd-sourcing. In this paper, we present a novel bias testing framework, BTC-SAM, which generates high-quality test cases for bias testing in SA models with minimal specification using Large Language Models (LLMs) for the controllable generation of test sentences. Our experiments show that relying on LLMs can provide high linguistic variation and diversity in the test sentences, thereby offering better test coverage compared to base prompting methods even for previously unseen biases.