So, with a bit of hacking and pokey points, I got SQLite using a custom filesystem in Rust. It “works” in the sense that, from no files, it generates a database that can be used by the SQLite CLI. It has some slightly rough parts still (it seems like later updates aren't persisted and the memory implementation segfaults because of some silly range coding) but this is promising!
A big thank you to everyone who's helped me reach my current fundraiser for support. I couldn't have done this without y'all.