Last time the similar question comes up there is no good solution: Updating Priors with Posterior for Vector-Valued Distributions, Updating priors for Gaussian Mixture Model
I think one thing worth to explore is to use some kind of approximation (MvNormal, for example). It would be an extension of: Performance speedup for updating posterior with new data and code here: https://github.com/junpenglao/Planet_Sakaar_Data_Science/blob/master/PyMC3QnA/Update%20prior%20with%20Interpolation.ipynb (see last cell)