I'm afraid that is not possible. The FLTK API does not detail the reason why a start_job() call may have failed.
That's mostly because that would be very OS-specific.
I have tried under MSWindows 10 the test/device.exe app and had it print to
"Microsoft print to pdf" twice, keeping open in Microsoft Edge the first resulting pdf file before
running the second print job.
The second print job fails at the StartDoc() OS call which returns -1, a sign of failure.
But Microsoft's doc gives no way to get more detail about the cause of that failure:
https://docs.microsoft.com/en-us/windows/win32/api/wingdi/nf-wingdi-startdoca
So, at least in that case, there's no way I can see to get more information than
"the print job failed to start".