Even for extremely idealized system like a Lennard-Jones fluid there is no analytical equation of state available, and only (very complicated) empirical expressions which are valid in limited pressure and temperature ranges are available (see for example here).

Sure, you can fit $p(T,\rho)$ with a complex enough polynomial or with some other function, but the resulting expression will be valid only in the range of the data from which you obtained the fit. Also, to obtain a good fit you would probably be using a lot of parameters; however, as John von Neumann once said: "With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.". Given enough parameters, you can fit any data set to any functional form, but very little physical insight is obtained from such a procedure (in other words, would you be able to give a physical interpretation of all the parameters you are using?).

In conclusion, my suggestion this: take data from the range of pressures and temperatures you are interested in and find whatever fits them best. You will thus have an empirical formula that works pretty well in that range (and only in that range!).

Even for extremely idealized system like a Lennard-Jones fluid there is no analytical equation of state available, and only (very complicated) empirical expressions which are valid in limited pressure and temperature ranges are available (see for example here).

Sure, you can fit $p(T,\rho)$ with a complex enough polynomial or with some other function, but the resulting expression will be valid only in the range of the data from which you obtained the fit. Also, to obtain a good fit you would probably be using a lot of parameters; however, as John von Neumann once said: "With four parameters I can fit an elephant, and with five I can make him wiggle his trunk". Given enough parameters, you can fit any data set to any functional form, but very little physical insight is obtained from such a procedure (in other words, would you be able to give a physical interpretation of all the parameters you are using?).

In conclusion, my suggestion is this: take data from the range of pressures and temperatures you are interested in and find whatever fits them best. You will thus have an empirical formula that works pretty well in that range (and only in that range!).