Contact Framework support, they should tell you if you’re covered: frame.work/contact-us.
Your purchase date of May 2023 does certainly look like it should be covered to me. Framework’s post about updated firmware and the warranty is dated Oct 2024, after your purchase.
They said the updated firmware was being released the same day, at least for one generation laptop. And that the extended warranty was for systems and batteries that shipped before the updated firmware was available. Which all sounds like it covers you.
If you need your laptop asap & can’t wait for an answer from support or a warranty replacement, you could buy a new battery from the marketplace. Though, I’m not sure if Framework has a procedure to refund you if they later come to the conclusion that your battery should be covered under warranty.
If you have older BIOS firmware then it could help for your next battery. So you should check. But it won’t help for your current swollen battery. I don’t believe swelling can be reversed.
Here is how to check your current BIOS version: How to check the BIOS version on Windows/Linux/BIOS.
You have a Framework Laptop 13, Intel Core 13th Gen, correct?