Many thanks @Stockyfor responding.
For other's benefit, I managed to travel from Bangkok to Hat Yai as a tourist with minimal hassle.
Nobody checked any of our COVID documentation (thailand pass, vaccination certificate) before, during or after the flight.
Arriving at the Hat Yai Airport we had to complete the Songkhla Care registration. This can be done online even beforehand, but it's asking for identity verification using the Line app. I don't have Line installed, but there were 3 terminals at the airport where a kind lady will help you fill out a similar form (but in Thai language) that will generate the magic QR code which lets you exit the airport.
Currently in Hat Yai, if anyone has any questions, happy to respond.