LBM for flow under different applied pressure

I want to test the flow of a fluid in a certain geometry under different pressure, say 10 Pa, 100 Pa, 1000 Pa and 10000 Pa to check the behavior of the flow under different pressure. All the other conditions remain constant.

But as far as I know, the LBM-BGK model uses ideal gas law, p = rho*c_s^2, and hence, it shows only small fluctuation in density, resulting in only a small fluctuation in pressure values. So, I think that this small fluctuation will be same regardless of what pressure is applied to the flow. So, am I right in thinking that there is no effect of applying different pressure on the flow as the fluctuation will be same and the fluid behavior will be same everywhere ? If no, then how can I capture the change in fluid behavior under different applied pressure ?