How to improve fit of Poisson Mixture Model in PyMC3?

As @adam said, you shouldnt model the data with mixture, as the data is time dependent, which means each data point is not iid.
In some cases if the cluster is well separated (i.e., large tau) or you have lots of data, you might still be able to do clustering using Mixture, but in this case you dont have enough data.
For multiple tau you can have a look at Switch point Metropolis tuning. But with number of tau unknown it will be difficult to deal with.