Highly pathogenic avian influenza (HPAI) is a contagious viral disease that has led to the culling of huge amounts of poultry as well as the mortality of countless wild birds. The recent panzootic that began in 2021 has been particularly notable for its devastating effect on seabird populations around the globe. Whilst transmission of HPAI within poultry has been relatively well studied, the recency of the current panzootic, combined with data collection challenges, means that much less is known about key epidemiological parameters, such as reproduction numbers, R0, of HPAI in wild populations. We develop methodology to carry out exact Bayesian parameter inference using reversible jump Markov chain Monte Carlo applied to mortality data in the form of daily carcass counts over the duration of subsequent outbreaks in a colony of common terns, Sterna hirundo, in 2022 and 2023. We estimate R0 to be 3.7 (95% CI 2.3; 7.2) in 2022, and 3.2 (95% CI 1.7; 7.0) in 2023. The probability of mortality for an infected bird was estimated to drop from 0.26 (95% CI 0.24; 0.28) in 2022 to 0.14 (95% CI 0.11; 0.20) in 2023. Our findings furthermore suggest direct bird-to-bird transmission to be the predominant driver of infection within the colony, with environmental transmission playing a negligible role.