Non-abelian topological codes are of great interest, since they provide the opportunity to achieve a universal fault-tolerant logical gate set through braiding, without the need for magic state distillation. So far results concerning thresholds of such ECCs have been obtained exclusively using phenomenological anyon models. Here, we present the first error-correction threshold for a microscopic model supporting anyons of which the braid group representation is universal. We do this by studying Turaev-Viro codes with the Fibonacci input category on a tailed hexagonal lattice. The code space is defined by a set of commuting projectors, related to the Levin-Wen plaquette and vertex operators. We introduce a set of measurements and unitary operators allowing one to take any state to the string-net subspace, and a set of measurement operators corresponding to the anyonic charge of a single plaquette. With tensor network techniques, we determine the action of Pauli noise on the string-net subspace, enabling us to reliably calculate measurement outcomes and state evolution. This in turn enables us to perform Monte-Carlo simulations of the Fibonacci LW code on a torus in order to obtain error thresholds. We study several decoders and compare their respective thresholds and performance.