Epidemic prevalence surveys monitor the spread of an infectious disease by regularly testing representative samples of a population for infection. State-of-the-art Bayesian approaches for analysing epidemic survey data were constructed independently and under pressure during the COVID-19 pandemic. In this paper, we compare two existing approaches (one leveraging Bayesian P-splines and the other approximate Gaussian processes) with a novel approach (leveraging a random walk and fit using sequential Monte Carlo) for smoothing and performing inference on epidemic survey data. We use our simpler approach to investigate the impact of survey design and underlying epidemic dynamics on the quality of estimates. We then incorporate these considerations into the existing approaches and compare all three on simulated data and on real-world data from the SARS-CoV-2 REACT-1 prevalence study in England. All three approaches, once appropriate considerations are made, produce similar estimates of infection prevalence; however, estimates of the growth rate and instantaneous reproduction number are more sensitive to underlying assumptions. Interactive notebooks applying all three approaches are also provided alongside recommendations on hyperparameter selection and other practical guidance, with some cases resulting in orders-of-magnitude faster runtime.