We prove large deviation principles (LDPs) for full chordal, radial, and multichordal SLE(0+) curves parameterized by capacity. The rate function is given by the appropriate variant of the Loewner energy. There are two key novelties in the present work. First, we strengthen the topology in the known chordal LDPs into the topology of full parameterized curves including all curve endpoints. We also obtain LDPs in the space of unparameterized curves. Second, we address the radial case, which requires in part different methods from the chordal case, due to the different topological setup. We establish our main results via proving an exponential tightness property and combining it with detailed curve escape probability estimates, in the spirit of exponentially good approximations in LDP theory. In the radial case, additional work is required to refine the estimates appearing in the literature. Notably, since we manage to prove a finite-time LDP in a better topology than in earlier literature, escape energy estimates follow as a consequence of the escape probability estimates.