We consider a zero-range process $η^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12Δu^α, α>1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing particle size $χ_N\to 0$. The key challenge is to develop uniform integrability estimate on the nonlinearity $(η^N(x))^α$ in a situation where neither pathwise regularity nor Dirichlet-form based regularity is readily available. We resolve this by introducing a novel multiscale argument exploiting the appearance of pathwise regularity across scales.