The issue seems to be that referral cookie attempts to set too late, after WooCommerce sessions have started or headers have been sent. On first visit, no session exists → cookie cannot be set → referral only works on second visit.

What I've tried to do is - launch referral logic at the earliest hooks - many times (even before posting this problem here) with different code snippets from ChatGPT, Claude, now also Lumo and Duck.ai (which, admittedly, also uses ChatGPT) but none of the solutions work. The end result is still the same.