Mastodon works just fine with Wasabi, Minio, Ceph and DigitalOcean Spaces, so it didn't even occur to me that there might be issues with another S3-compatible API. I've contacted Exoscale support. I feel incredibly foolish for not having sorted this out before beginning a costly move.
Figured out which configuration option was wrong thanks to one of Flexify's engineers. Gonna have to add an extra environment variable for this edge case 😩