the bridge plugin is meant to be installed as a plugin. However, copying it into your functions.php does give you the opportunity to directly manipulate it.
the default WooCommerce shop loop archive shows the title on the woocommerce_shop_loop_item_title
hook with the default (10) priority.
/**
* Hook: woocommerce_shop_loop_item_title.
*
* @hooked woocommerce_template_loop_product_title - 10
*/
do_action( 'woocommerce_shop_loop_item_title' );
`
In my bridge plugin, I have added the subtitle to this same hook with a priority of 20, https://github.com/helgatheviking/kia-subtitle-woocommerce-bridge/blob/master/kia-subtitle-woocommerce-bridge.php#L30
So it should be displaying after the title.
Therefore, I would have to assume that your theme is doing something different with the main title? It’s impossible for me to say since I don’t know your theme.
You could try changing that 20 to a higher number, but if your theme has moved the title to a different hook it may not have any impact.