Part of the answer is here [Difference between different types of moments)
There are no general rules as to whether user theoretical or simulated moments. I prefer theoretical ones. Regarding the filtering: there are also no general guidelines. Best practice often seems to be to treat your model and data variables equally. That is, if you apply a Baxter-King filter to the data, do so as well for the model variables. In this case, as there are no theoretical moments available for the Baxter-King filter, you need to use simulated series.