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A wide variety of battery models are available, and it is not always obvious which model `best' describes a dataset. This paper presents a Bayesian model selection approach using Bayesian quadrature. The model evidence is adopted as the selection metric, choosing the simplest model that describes the data, in the spirit of Occam's razor. However, estimating this requires integral computations over parameter space, which is usually prohibitively expensive. Bayesian quadrature offers sample-efficient integration via model-based inference that minimises the number of battery model evaluations. The posterior distribution of model parameters can also be inferred as a byproduct without further computation. Here, the simplest lithium-ion battery models, equivalent circuit models, were used to analyse the sensitivity of the selection criterion to given different datasets and model configurations. We show that popular model selection criteria, such as root-mean-square error and Bayesian information criterion, can fail to select a parsimonious model in the case of a multimodal posterior. The model evidence can spot the optimal model in such cases, simultaneously providing the variance of the evidence inference itself as an indication of confidence. We also show that Bayesian quadrature can compute the evidence faster than popular Monte Carlo based solvers.