We present observations, numerical simulations, and analysis from experiments in the Lithium Tokamak Experiment-Beta (LTX-β) in which the electron temperature profile (Te
(r)) shifts from flat to peaked and a tearing mode is also destabilized when the average density (ne
ave
) exceeds ~1019 m-3. Flat Te
(r) is obtained routinely in LTX-β, with a lithium coated, low-recycling first wall, once the external fueling is stopped and density decays [D Boyle et al., Nucl. Fusion 63 (2023) 056020]. In the present experiment, flat Te
(r) can be sustained while maintaining constant ne
ave
below a line averaged density threshold (ne
ave
th
) of ~ 1019 m-3. Above ne
ave
th
, Te
(r) shifts from flat to peaked and a tearing mode is destabilized. Due to low recycling, the achieved ne
ave
can be controlled precisely by external fueling and hence, a certain threshold of the edge neutral inventory from the external fueling is experimentally manifested through ne
ave
th
. The goal of the present work is to investigate the role of edge neutrals in determining Te
(r) and MHD stability in the unique low-recycling regime of LTX-β. Our hypothesis is that the peaking of Te
(r) beyond ne
ave
th
is due ultimately to the edge cooling by the cold neutrals beyond a critical fueling flux. At lower fueling flux, flat Te
(r) results in broader pressure profile and lower resistivity, which in turn stabilizes the tearing mode. This hypothesis is supported by edge neutral density estimation by DEGAS 2 code. Mode analysis by singular value decomposition confirms the tearing mode structure to be m/n = 2/1 (m and n being the poloidal and toroidal mode numbers). Linear tearing stability analysis with M3D-C1 predicts that plasmas with ne
ave
> 1019 m-3 are highly susceptible to a n = 1 tearing mode. ORBIT simulations, however, confirmed that the tearing modes do not contribute to the loss of fast ions from neutral beam injection. This study shows for the first time that the neutral inventory at the edge could be one of the deciding factors for the achievability of the unique operation regime of flat Te
(r) and the excitation of tearing activity that could be disruptive for the plasmas.