Is there a way to create referral links and monitor the subscriptions to the product based on who referred the patron?
If not, do you have any creative ideas to use Patreon and still be able to do similar things? (I have no problem with writing code and using the API if needed, the only question is whether it’s possible…)
Directly via Patreon, unlikely. However using the WP plugin and integrating it with an affiliate plugin, you can drop an affiliate cookie to visitors who are not patrons, and if the user signs up via an affiliate link, you can get that affiliate cookie when the user visits your site and credit that affiliate after checking if the patron signed up in the allowed duration for affiliate credit.