The spatial logistic branching process is a population dynamics model in which particles move on a lattice according to independent simple symmetric random walks, each particle splits into a random number of individuals at rate one, and pairs of particles at the same location compete at rate c. We consider the weak competition regime in which c tends to zero, corresponding to a local carrying capacity tending to infinity like 1/c. We show that the hydrodynamic limit of the spatial logistic branching process is given by the Fisher-Kolmogorov-Petrovsky-Piskunov equation. We then prove that its non-equilibrium fluctuations converge to a generalised Ornstein-Uhlenbeck process with deterministic but heterogeneous coefficients. The proofs rely on an adaptation of the method of v-functions developed in Boldrighini et al. 1992. An intermediate result of independent interest shows how the tail of the offspring distribution and the precise regime in which c tends to zero affect the convergence rate of the expected population size of the spatial logistic branching process to the hydrodynamic limit.