Identification of optically active defects in hexagonal boron nitride (hBN) remains challenging because different defect candidates can produce similar calculated photoluminescence (PL) lineshapes. Here, we demonstrate that force-constant embedding methodology can be extended to calculate high-resolution, vibrationally resolved PL lineshapes in the polar layered material hBN. Benchmarking against the carbon dimer defect, responsible for the 4.1 eV emission, we show that the methodology accurately reproduces the low-energy acoustic phonons and fine spectral features that are inaccessible when using direct supercell calculations. Applying the methodology to neutral carbon trimers (C₂ C_B and C₂ C_N), we find that the calculated PL lineshape and excitation energy of C₂ C_N closely match the low-temperature spectrum of the emitter with a zero-phonon line at 1.54 eV. These results establish high-resolution phonon-sideband modeling as a powerful tool for discriminating among candidate defects in polar materials.