The 'bereavement policy' is offered at SQ's discretion, so the fact that they're willing to give you a partial refund is more than good enough.

Your prior experience with Southwest is moot.

IIRC when a friend had to do this, SQ simply required the death certificate. I'd assume this hullabaloo is entirely due to Budget Air not knowing what they're doing.

Legally, you don't have a leg to stand on - especially if you can't even get your stories straight here, let alone in a court of law.

If I were you I'd insist on Budget Air forwarding your documents to SQ and asking for proof since it sounds like they haven't even bothered to do so (for fear of rejection??), and accepting the partial refund, whatever it comes to.