We present a Bayesian analysis of the distances to 15,040 Large Magellanic Cloud (LMC) RR Lyrae stars using $V$- and $I$-band light curves from the Optical Gravitational Lensing Experiment, in combination with new $z$-band observations from the Dark Energy Camera. Our median individual RR Lyrae distance statistical error is 1.89 kpc (fractional distance error of 3.76 per cent). We present three-dimensional contour plots of the number density of LMC RR Lyrae stars and measure a distance to the core LMC RR Lyrae centre of $50.2482±0.0546 m̊(statistical) ±0.4628 (̊systematic) rk̊pc$, equivalently $μ_rmM̊C=18.5056±0.0024 rm(åtistical) ±0.02 rm(st̊ematic)$. This finding is statistically consistent with and four times more precise than the canonical value determined by a recent meta-analysis of 233 separate LMC distance determinations. We also measure a maximum tilt angle of $11.84^∘±0.80^∘$ at a position angle of $62^∘$, and report highly precise constraints on the $V$, $I$, and $z$ RR Lyrae period–magnitude relations. The full dataset of observed mean-flux magnitudes, derived colour excess $E(V-I)$ values, and fitted distances for the 15,040 RR Lyrae stars produced through this work is made available through the publication's associated online data.