Welcome to WP StackExchange! You may want to try something like this, which uses the same rewrite setting as your register CPT:
$args = array(
'rewrite' => array('slug' => 'events/ecat', 'with_front' => false)
);
register_taxonomy( 'ecat', array( 'events' ), $args );
Sources:
https://cnpagency.com/blog/the-right-way-to-do-wordpress-custom-taxonomy-rewrites/
https://codex.wordpress.org/Function_Reference/register_taxonomy#Arguments
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