Turns out that there's no way to download past invoices anywhere. The button to manage the subscription simply disappears, instead being replaced by a button to "Renew Plus".
I even found an old link to the Stripe page where the invoices are listed, but it looks like the previous invoices were simply deleted.
Looks like I gotta pay 23€ to access old invoices. Who decided that was a good (read: legal) idea? The vibe coding intern??