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Fluctuations in Lyman-$α$ (Ly$α$) forest transmission towards high-$z$ quasars are partially sourced from spatial fluctuations in the ultraviolet background (UVB), the level of which are set by the mean free path of ionizing photons ($λ_{\text{mfp}}$). The auto-correlation function of Ly$α$ forest flux characterizes the strength and scale of transmission fluctuations and, as we show, is thus sensitive to $λ_{\text{mfp}}$. Recent measurements at $z \sim 6$ suggest a rapid evolution of $λ_{\text{mfp}}$ at $z>5.0$ which would leave a signature in the evolution of the auto-correlation function. For this forecast, we model mock Ly$α$ forest data with properties similar to the XQR-30 extended data set at $5.4 \leq z \leq 6.0$. At each $z$ we investigate 100 mock data sets and an ideal case where mock data matches model values of the auto-correlation function. For ideal data with $λ_{\text{mfp}}=9.0$ cMpc at $z=6.0$, we recover $λ_{\text{mfp}}=12^{+6}_{-3}$ cMpc. This precision is comparable to direct measurements of $λ_{\text{mfp}}$ from the stacking of quasar spectra beyond the Lyman limit. Hypothetical high-resolution data leads to a $\sim40\%$ reduction in the error bars over all $z$. The distribution of mock values of the auto-correlation function in this work is highly non-Gaussian for high-$z$, which should caution work with other statistics of the high-$z$ Ly$α$ forest against making this assumption. We use a rigorous statistical method to pass an inference test, however future work on non-Gaussian methods will enable higher precision measurements.