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Photonic bound states in the continuum (BICs) are a standout nanophotonic platform for strong light-matter coupling with transition metal dichalcogenides (TMDCs), but have so far mostly been employed as all-dielectric metasurfaces with adjacent TMDC layers, incurring limitations related to strain, mode overlap, and material integration. In this work, we experimentally demonstrate for the first time asymmetry-dependent BIC resonances in 2D arrays of monolithic metasurfaces composed solely of the nanostructured bulk TMDC WS$_2$ with BIC modes exhibiting sharp and tailored linewidths, ideal for selectively enhancing light-matter interactions. Geometrical variation enables the tuning of the BIC resonances across the exciton resonance in bulk WS$_2$, revealing the strong-coupling regime with an anti-crossing pattern and a Rabi splitting of 116 meV. The precise control over the radiative loss channel provided by the BIC concept is harnessed to tailor the Rabi splitting via a geometrical asymmetry parameter of the metasurface. Crucially, the coupling strength itself can be controlled and is shown to be independent of material-intrinsic losses. Our BIC-driven monolithic metasurface platform can readily incorporate other TMDCs or excitonic materials to deliver previously unavailable fundamental insights and practical device concepts for polaritonic applications.