Typically, the UI and jobs sides need shared access to a common database, so SQLite generally won’t work in this configuration.
(Possibly I’ve incorrectly guessed what you’re doing, though.)
I.e., whatever volume the UI Machine has won’t be accessible from the jobs Machine, and vice versa.
Moreover, having only a single volume and having only a single UI Machine both come with major caveats on the Fly.io platform, although @rubys has been working on mitigating the former.
There was talk back in October about making the SolidQueue+SQLite combination more tenable on Fly.io, but I don’t know how far that’s progressed…
(Also, one user reported getting SolidQueue+LiteFS working, which is a lot of effort, but was a definite achievement that does solve both.)