Context. In planet-forming disks, deuterated species like DCO⁺ often show up in rings. Two chemical formation routes contribute: cold deuteration at temperatures below 30 K and warm deuteration at temperatures up to 80 K. Aims. We aim to reproduce the DCO⁺ emission in the disk around HD 163296 using a simple 2D chemical model for the formation of DCO⁺ through the cold deuteration channel and a parametric treatment of the warm deuteration channel. Methods. We use data from ALMA in band 6 to obtain a resolved spectral imaging data cube of the DCO⁺ J = 3–2 line in HD 163296 with a synthesized beam of 0.′′53 × 0.′′42. We adopt a physical structure of the disk from the literature that reproduces the spectral energy distribution. We then apply a simplified chemical network for the formation of DCO⁺ that uses the physical structure of the disk as parameters along with a CO abundance profile, a constant HD abundance, and a constant ionization rate. We model the contribution of the warm deuteration channel with two parameters: an effective activation temperature and a constant abundance. Finally, from the resulting DCO⁺ abundances, we calculate the non-LTE emission using the 3D radiative transfer code LIME. Results. The observed DCO⁺ emission is reproduced by a model with cold deuteration producing abundances up to 1.6 × 10⁻¹¹. Warm deuteration, at a constant abundance of 3.2 × 10⁻¹², becomes fully effective below 32 K and tapers off at higher temperatures, reproducing the lack of DCO⁺ inside 90 AU. Throughout the DCO⁺ emitting zone a CO abundance of 2 × 10⁻⁷ is found, with ~99% of it frozen out below 19 K. At radii where both cold and warm deuteration are active, warm deuteration contributes up to 20% of DCO⁺, consistent with detailed chemical models. The decrease in DCO⁺ at large radii is attributed to a temperature inversion at 250 AU, which raises temperatures above values where cold deuteration operates. Increased photodesorption may also limit the radial extent of DCO⁺. The corresponding return of the DCO⁺ layer to the midplane, together with a radially increasing ionization fraction, reproduces the local DCO⁺ emission maximum at ~260 AU. Conclusions. We can successfully reproduce the observed morphology of DCO⁺ at large radii by only considering the dependence on temperature in the chemical reactions that produce it. Predictions on the location of DCO⁺ within the disk from simple models depend strongly on the gas temperature. Outer disk temperature inversions, expected when grains decouple from the gas and drift inward, can lead to secondary maxima in DCO⁺ emission and a reduction of its radial extent. This can appear as an outer emission ring, and can be used to identify a second CO desorption front.