Per NetWallah's reply, I changed my code to this:

$self->{db} = DBI->connect(
    "dbi:SQLite:dbname=$db_file",
    "",
    "",
    {RaiseError => $self->{db_err}}
) or die $DBI::errstr;
[download]

...and my tests to send in db_err => 0 in the calls to new(), and I get 100% coverage. I don't think I'll make this a user-visible param (but I will document it) as I always want the upper module to take care of the problem, but to ensure I've covered it, it works.

This was a case of me not reading enough documentation.