Hi,
Thanks for using our plugins I can provide some information on this.
The Filter Bar is using the query string of the url (?tribe_eventcategory[]=53 ) to filter the Calendar. It is required to be a query string and changing the url to a pretty permalink would break it. So I am afraid it is not possible to change the url to the way you would like for the filter bar.
There are category views if you would like to use those, for instance the Dog Adoption Events Category has a url of this:
http://www.seattledogspot.com/events/category/dog-adoption-events/
Let me know if you have any follow up questions.
Thanks