<p>In the framework of Brans–Dicke(BD) theory, we have considered a Bianchi type (BT)–III anisotropic spacetime with a cosmological model consisting of Sharma-Mittal Holographic Dark Energy (SMHDE) and pressureless matter. We analyse the temporal evolution of important parameters for the deceleration parameter, the Equation of State (EoS) parameter, state finder diagnostics, skewness, scalar field, and the squared speed of sound. We observe a smooth transition from an early decelerating epoch to a late accelerating epoch. The EoS parameter evolves from a quintessence-like phase to a phantom regime at late times. Moreover, the scalar field and matter density show negligible values at the early times and increase dramatically towards later times with the role of Sharma-Mittal entropy corrections. Compared with the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Lambda\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>CDM model, the SMHDE setup incorporates the main behaviour of late-time acceleration into the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Lambda\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>CDM model and, in addition shows dynamic dark energy, phantom crossing, and entropy-related correction. These results indicate that this SMHDE model in BD gravity is theoretically extended and observationally viable as compared to the standard <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Lambda\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>CDM cosmology. Additionally, we investigate the classical stability of the models by analysing the squared sound speed parameter.</p>