You can look a bit more towards breaking it down into more advanced detail and the mathematics of hydrostatic and how it fails in Doswell and Markowski's paper on buoyancy.

But, with such limited model output, not sure that you're looking to get into all of that detail, as I don't believe you'll find any extra help going those avenues with the variables you have.

With just slightly more information, most importantly temperature, you could use the ideal gas law:

 p = ρRT 

^{[where R is the ideal gas constant of the gas under consideration and T is temperature in Kelvin]}

This would allow you replace the unknown density, and you'd then end up with:

w = −ωRT/pg

Which you could then either further approximate R to be that for dry air = Rd = 287 J/Kkg, or even work up the adjustment your moisture content variable causes (though don't think doing so would offer much useful improvement as you've already got greater errors due to the failings of the hydrostatic approximation).

Since you don't have temperature, it seems your only remaining option would be to plug in the density approximation for the level of interest from the approximated "Standard Atmosphere" here (use the table where ρ is in units of kg/m^3).

It has been a long while since I've done this stuff, but I don't see any better options. Unless somehow there's a way to work from the geopotential, maybe apply something like mass continuity in pressure coordinates, and get to something more complex to return something more correct for w. But if there is, it outpaces me!

So, long story short, it appears that your best hope is w = −ωRT/pg, but it won't be 100% reliable, particularly in strong convection.