I think this could be done with Crumbs programmatically, but this might be overkill. It would also be fragile, because it would depend on Crumbs functionality that is not really part of the public API, so could change in future versions of Crumbs.
Also consider that you don't really need the full breadcrumb, just the taxonomy trail.
Instead, you could write a custom Views header plugin. Unfortunately it is a bit hard to find documentation on that.
Or, you could alter the views header with one of the views hooks.
Or you could use the theme layer to override the views theme output for this specific view.
Or override the block template for the specific block. I think this one is actually the easiest for a starter, and for this specific situation. You can get the taxonomy term from the node, and you get the node with menu_get_object().
Have a look at https://api.drupal.org/api/drupal/modules%21taxonomy%21taxonomy.pages.inc/function/taxonomy_term_page/7 to see how the taxonomy page calculates a breadcrumb based on the taxonomy hierarchy.
(I am mentioning more than one solution here, because other readers might have slightly different requirements.)