Given only pressure and potential equivalent temperature, can I determine other variables?
I think you cannot, might be wrong on that though.

In a single-fluid atmosphere without condensibles, the pressure is essentially proportional to $P \sim \rho \, T$ and potential temperature is essentially $\theta \sim T 
P^{1-1/\gamma}$, with the temperature $T$, and the adiabatic index of the gas $\gamma$.
So in this situation you could reconstruct density and temperature from $P$ and $\theta$, but if you add another fluid into the mix (which is your condensible), then I don't see how that should be possible.