How should one assess the best fitting model in case of overdispersion and/or serial correlation?

Unfortunately, the Goodness-of-fit tests are not valid if the counts are not independent Poisson observations. This hampers the evaluation of the fit of the model in case of substantial overdispersion (say, > 3) and serial correlation (say, > 0.4). In such cases, you have to rely on the Wald tests to find the best model.