Part of the answer turned out to be that their website was Flash, so it was never meaningfully indexed, and their MySpace was never mirrored. A band can be entirely findable in 2006 and entirely gone by 2026 depending on which technologies they happened to pick.
It was eventually solved by searching on the producer's credits rather than the band's name, and then by two people replying to cold emails. Both the producer and the singer were generous enough to be interviewed, so there's some fresh detail in there about the recording session as well.