What are typical values for the proportionality constant of the barotropic equation of state in fluid

I'm trying to create a fluid simulation, where I'm working with a barotropic equation of state for simplicity. Assuming I have some fluid with a barotropic equation of state, e.g. a very simple atmosphere:

$$p = \rho \cdot A$$ Where $p$ is the pressure in $\rm kg/m/s^2$, $\rho$ is the density in $\rm kg/m^3$ and $A$ is some proportionality constant in ($\rm m^2/s^2$).

For air, I can find a value for $A$ by dividing pressure by density at sea-level, for example, but within this approximation, how would I find values that represent oil or water? I've looked at a few papers on atmospheric simulations, but none of them really explicitly deal with the value of $A$.

I've been somewhat successful in creating a simulation where "water" floats on "oil" (basically 2 fluids with a different (unrealistic?) values of $A$), but I'm trying to recreate this with more appropriate values.

To the extent that a barotropic equation of state is realistic for these fluids, you can use the same idea as you proposed for air. That is, find the density of each fluid at some pressure and divide the two. Suitable reference conditions might be STP or room temperature, at which the density of many substances is known. For instance, here is a paper studying the densities of various cooking oils as functions of temperature. Almost all measurements for the oils range from $850\,\mathrm{kg/m^3}$ (at high temperature) and $920 \,\mathrm{kg/m^3}$ (at room temperature). These measurements seem to have been made at atmospheric pressure, around $10^5\,\frac{\mathrm{kg}}{\mathrm{m\cdot s^2}}$. So depending on the typical temperature you expect your simulation to run at, you could reasonably estimate $A$ for vegetable oil to lie around $0.0085-0.0092\,\mathrm{m^2/s^2}$. Likewise, for water at room temperature and atmospheric pressure, you would get $A\approx \frac{998\,\mathrm{kg/m^3}}{10^5\,\mathrm{kg/m^2\cdot s^2}} \approx 0.00998\,\mathrm{m^2/s^2}$.