Connect Content Type with Related Taxonomy Fields using Template
I solved this by creating two new regions in my theme underneath Content, then I created a View with two blocks: one for taxonomy B and one for taxonomy C. 

The view pulls in taxonomy fields using the relationship: Content: Taxonomy terms on node. 

I have setup the URL pattern for my content type to be contentTypeA/taxonomyB/taxonomyC. By doing this, I was able to reference the correct term by using a Contextual Filter: Content: Has taxonomy term ID. 

The Contextual Filter was a little tricky for me. As it was coming from the URL at a specific level of the path, I found that I could access taxonomyB through the path component 2, and taxonomyC through path component 3. 



Under WHEN THE FILTER VALUE IS AVAILABLE OR A DEFAULT IS PROVIDED I used Specify validation criteria as the respective taxonomy term. 



Finally, using Term name converted to Term ID allowed it to see the plain english term name in the URL path.