Set shipping method programmatically

I just created a custom shipping method with the id of custom_shipping. The option now appears on the Woocommerce -> Settings -> Shipping admin page.

I would like to set the shipping method dynamically based on a user’s credentials. The default method is flat_rate right now.

Question: How can I set the shipping method to custom_shipping for users who meet a requirement, for the entirety of their session?

Tried:
$woocommerce->session->set('chosen_shipping_methods', array('purolator_shipping');

This sets the session variable chosen_shipping_methods correctly on my cart page, but upon moving to checkout, the session goes back to using the flat_rate shipping method.

There must be a hook or filter that I can plug into to change the shipping method upon cart-session creation or something. (When a user adds something to cart for first time).

I would ideally like to set the new shipping method before anything else is loaded so the shipping method looks correct on their cart and checkout summaries.

Guidance appreciated.

https://ww.wp.xz.cn/plugins/woocommerce/