Why am I not seeing the CoCart-API-Cart-Key header returning?

If CoCart-API-Cart-Key header is not returning then you need to enable CORS so that the header is exposed.

Either enable CORS support via a filter shown below or simply install CoCart CORS plugin for quicker activation.

That’s it.

<?php
add_filter( 'cocart_disable_all_cors', function() { return false; });