Motivated by the low-energy effective action of heterotic string theory, where the Kalb-Ramond (KR) two-form and nonlinear gauge corrections arise simultaneously, we investigate a static, spherically symmetric black hole in Einstein gravity coupled to a KR field and ModMax nonlinear electrodynamics. The solution depends, beyond mass and charge, on the Lorentz-symmetry-breaking parameter $\ell$, the ModMax deformation parameter $\gamma$, and a discrete branch selector $\zeta =\pm 1$. We show that the ordinary branch admits extremal and nonextremal configurations, while the phantom branch generically supports a single-horizon geometry. Black hole thermodynamics is analyzed within the Tsallis nonextensive framework, revealing branch-dependent stability and Joule-Thomson behavior. Weak gravitational lensing is computed via the Ono-Ishihara-Asada extension of the Gauss-Bonnet theorem, yielding a negative topological correction that reduces light bending relative to the Schwarzschild baseline—opposite in sign to Barriola-Vilenkin monopole backgrounds. Photon-sphere properties in plasma environments and tidal forces through geodesic deviation are also studied, revealing a universal tidal balance ratio $R_{\mathrm{rad}}/R_{\mathrm{ang}}=3/2$ in the ordinary branch. These multichannel signatures provide concrete observational handles for constraining the KR-ModMax framework through Event Horizon Telescope data and next-generation interferometric arrays.

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• Alternative gravity theories
• Classical black holes