One well-known approach is exactly what you describe (if I understood correctly): i.e. combine the inferences from the analyses of a large number of imputed datasets (each analyzed separately) by just using the equally weighted mixture distribution of the posterior distributions as the combined posterior distribution. In practice, this means that we just throw together all the (or a random subset) of the MCMC samples that we generated by analyzing each of the multiply imputed datasets on their own. We use this combined set of MCMC samples from all the models, as if they were a giant single set of MCMC samples. This is described in Bayesian Data Analysis (3rd edition by Gelman et al. 2014 on page 452). It's also investigated and reported to perform well with a sufficiently large number of imputations (e.g. 100) by Zhou and Reiter 2010 (Zhou, X. and Reiter, J.P. 2010. A note on Bayesian inference after multiple imputation. The American Statistician, 64(2), pp. 159-163.).
Note: checks for non-convergence need to be done within each imputed datasets. E.g. if you treat chains from separate imputations as if they were separate chains run on the same datasets, checks like Rhat etc. will tell you that you have a problem, because - unsurprisingly - differently imputed datasets do result in (slightly) different posterior distributions. With that caveat, I've essentially treated them like separate chains (e.g. if I had 4 chains per imputation and 100 imputation, I thereafter proceeded as if I had 400 chains).
There's alternatives to this. E.g. it is also clear that you could also run a single MCMC sampler across all imputations and at each MCMC iteration use the "mixture-likelihood" for all the imputations, but I'm not aware that any software does this easily (you can of course hand-code this in something like Stan). The alternative to have a single model that imputes and analyzes at the same time is of course also a serious option, but again harder to implement. Keeping imputation and analysis as separate steps also has advantages in terms of addressing different questions/estimands (e.g. jump-to-reference-group imputation becomes easier to implement).