Does zero-lift angle of attack depend on Reynolds number? In this post I read that, due to viscous effects, the $C_{l}$ at a given angle of attack increases slightly with the Reynolds number but it seems that it just happens near the stall angle of attack.

I am performing some wind tunnel experiments at much lower Reynolds than those of the figure of the linked post (150,000 and 300,000) with the same profile, NACA 4412. My question is if at the mentioned Reynolds, can there be a small variation in the angle at which $C_{l}=0$, and therefore my $C_{l}-\alpha$ curves at Re=150,000 be noticeably below the Re=300,000 ones.
The aerodynamic coefficients  $c_l,c_d,c_m$ are in general functions of the angle of attack $\alpha$, Reynolds $Re$ and Mach number $Ma$. For your experiments, since you are operating in a very low $Re$ regime, assuming incompressible flow, your lift, drag, moment curves will largely depend on viscous phenomena (i.e. $Re$ number). Generally, as $Re$ number drops, the boundary layer thickens. For this reason, you will expect a lower $c_l$ number (due to viscous decambering) and a higher $c_d$.

You may also discover non-linear behaviors even on operating angles due to laminar separation bubble forming/bursting on the suction surface and this is very dependent on the turbulent intensity of your wind tunnel so the stall angles and post-stall behavior may not be the same for flight conditions where turbulent intensity is very low.

Finally, you can confirm with Xfoil that for $Re=300k \Rightarrow \alpha_{L=0}=-4.33\ deg$ while for $Re=150k \Rightarrow \alpha_{L=0}=-3.64\ deg$ for the NACA4412.