The absolute theoretical maximum mass that 1,600,000 lbs of powder could launch to the moon is a bit under 35,500 kg. The calculation is not all that difficult (introductory calculus-based physics), but is somewhat long and involved, and would be quite ugly on a site like this without mathjax.
However, that makes two major assumptions that cannot actually happen: no air resistance, and all of the powder burns instantly, transferring all its energy to the rocket (none to a visual flame, none to sound, etc. I'm still working on a quantitative analysis of these effects, but I'm quite certain that accounting for either air resistance or the finite burn speed of the rocket would make it impossible to reach the moon.
The maximum speed a propellant-powered rocket can reach depends on the rocket's mass, the mass of propellant, and the exhaust velocity of the exhaust. As per James Jenkins, the ship was 20,000 lbs; the propellant is, of course, 1,600,000 lbs. Using a typical black powder exhaust velocity of 800 m/s. Without fighting gravity, that amount of gunpowder could propel the ship to a bit over 3,500 m/s, well short of the Earth's escape velocity of 11,200 m/s. Turning it around a bit, that mass ratio would require an exhaust velocity of nearly 2,550 m/s. And for completeness, the given 160M lbs of propellant could launch a rocket of a bit under 1.5 lbs; it would take 24 billion pounds of propellant to launch the full 20,000-lb rocket.