Count data that are subject to both under and overdispersion at some hierarchical level cannot be readily accommodated by classic models such as Poisson or negative binomial regression models. The mean-parameterised Conway–Maxwell–Poisson distribution allows for both types of dispersion within the same model, but is doubly intractable with an embedded normalising constant. We propose a look-up method where pre-computing values of the rate parameter dramatically reduces computing times and renders the proposed model a practicable alternative when faced with such bidispersed data. The approach is demonstrated and verified using a simulation study and applied to three datasets: an underdispersed small dataset on takeover bids, a medium dataset on yellow cards issued by referees in the English Premier League prior to and during the Covid-19 pandemic, and a large Test match cricket bowling dataset, the latter two of which each exhibit over and underdispersion at the individual level.