Tiny Core Linux seems to be 32-bit by default (inferred from their downloads page which links to a separate 64-bit version)
Their minimum package is GUI-less.
Haven't used it myself, but maybe it's what you're looking for. Spin it up in a VM, etc.
This site lists that and four others: https://www.tecmint.com/lightweight-linux-distros-without-gui/ (I only looked at Tiny Core for the sake of this comment, so I don't know if the others are 32-bit or not.)
Crazy alternative: FreeDOS. Pros: Lots of abandonware out there for that platform. Cons: I can't tell how good its support is for USB-connected drives and networking. It definitely has some. Maybe it would be enough for your hardware.