The star β Pictoris is widely known for harbouring a large population of exocomets, which create variable absorption signatures in the stellar spectrum as they transit the star. Although the physical and chemical properties of these objects have long been elusive, the recently developed exocomet curve of growth technique has, for the first time, enabled estimates of exocometary column densities and excitation temperatures, based on absorption measurements in mutliple spectral lines. Using this new tool, we present a refined study of a β Pic exocomet observed on December 6, 1997 with the Hubble Space Telescope. We first show that the comet’s signature in Fe II lines is well explained by the transit of two gaseous components, with different covering factors and opacities. Then, we show that the studied comet is detected in the lines of other species, such as Ni II and Cr II. These species are shown to experience similar physical conditions as Fe II (same radial velocity profiles and same excitation temperatures), hinting that they are well mixed. Finally, using almost 100 Fe II lines rising from energy levels between 0 and 33 000 cm–1, we derive the complete excitation diagram of Fe+ in the comet. The transiting gas is found to be populated at an excitation temperature of 8190 ± 160 K, very close to the stellar effective temperature (8052 K). Using a model of radiative and collisional excitation, we show that the observed excitation diagram is compatible with a radiative regime, associated with a close transit distance (≤60 R⋆ ∼ 0.43 au) and a low electronic density (≤107 cm–3). In this regime, the excitation of Fe+ is controlled by the stellar flux, and does not depend on the local electronic temperature or density. These results allow us to derive the Ni+/Fe+ and Cr+/Fe+ ratios in the December 6, 1997 comet, at 8.5 ± 0.8 ⋅ 10–2 and 1.04 ± 0.15 ⋅10–2, respectively, close to solar abundances.