Basalt fiber reinforced recycled aggregate concrete (BFRAC) is a high-performance, environmentally friendly material that combines lightweight, high-strength fibers with low-carbon recycled aggregates (RAs), positioned for extensive use in building structures. However, research on its constitutive relationships is currently scarce, which partly restricts component design and analysis. In this context, the current study thoroughly explores the stress–strain relationship and variability of BFRAC under compression, using 240 cylinders for testing to investigate the influence of factors like coarse/fine RA sources, RA replacement rates, and fiber dosage. The study found that the addition of RAs and fibers reduced the workability of the mixture, particularly with the inclusion of fine RAs and short-cut fibers. Using coarse and fine RAs generally reduces the material’s elastic modulus, compressive strength, and post-peak ductility. Adding fibers can slightly improve compressive strength and peak strain, significantly reduce material brittleness, and have a minimal impact on elastic modulus. Importantly, the study noted that the pre-peak segment of the stress–strain curve of BFRAC is most sensitive to the addition of fine RAs, while the post-peak segment is most sensitive to fiber content. Despite this, using high-quality RAs up to 50% replacement and adding 0.4% by volume of fiber can make BFRAC with mechanical properties comparable to natural aggregate concrete. Based on the observed tests, this paper proposes constitutive relationships that incorporate skeleton curves and variability at different points for the compressive stress–strain behavior of BFRAC.