Household outbreak data are often associated with missing values in both outcomes and risk factors. This issue is further complicated by the fact that both source and time of transmission are not observable. In a typical epidemic, we only observe symptom onset times, partial test results, and possibly clustering structure such as households. Here we propose to analyze household-based transmission networks using a continuous-time Bayesian MCMC framework. We aim to devise an efficient sampling approach to handle the high-dimensional missing and latent data. With regard to the unobserved transmission network, we compare two sampling schemes, one using a marginalized likelihood that integrates out the uncertainty about potential infectors for each infection, and the other sampling the infectors using auxiliary data such as genetic sequences. We further include covariates for susceptibility & infectivity in our model and analyze the data using Accelerated Failure Time model. These methods are validated in simulation studies and will be applied to household COVID-19 transmission data.