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We study the sensitivity of the $z=0.1$ Lyman-$α$ Forest observables, such as the column density distribution function (CDD), flux PDF, flux power spectrum, and line width distribution, to sub-grid models of active galactic nuclei (AGN) feedback using the Illustris and IllustrisTNG (TNG) cosmological simulations. The two simulations share an identical Ultraviolet Background (UVB) prescription and similar cosmological parameters, but TNG features an entirely reworked AGN feedback model. Due to changes in the AGN radio mode model, the original Illustris simulations have a factor of 2-3 fewer Lyman-$α$ absorbers than TNG at column densities $N_{\rm HI}< 10^{15.5}$ cm$^{-2}$. We compare the simulated forest statistics to UV data from the Cosmic Origins Spectrograph (COS) and find that neither simulation can reproduce the slope of the absorber distribution. Both Illustris and TNG also produce significantly smaller line width distributions than observed in the COS data. We show that TNG is in much better agreement with the observed $z=0.1$ flux power spectrum than Illustris. We explore which statistics can disentangle the effects of AGN feedback from alternative UVB models by rescaling the UVB of Illustris to produce a CDD match to TNG. While this UVB rescaling is degenerate with the effect of AGN feedback on the CDD, the amplitude and shape of the flux PDF and 1D flux power spectrum change in a way distinct from a scaling of the UVB. Our study suggests that the $z=0.1$ Lyman-$α$ forest observables can be used as a diagnostic of AGN feedback models.