Hi again, have you tried the documented way to access the DBI error, by using a hook?

database_error

Called when a database error is raised by DBI. Receives two parameters: the error message being returned by DBI, and the database handle in question.

(See https://metacpan.org/pod/Dancer2::Plugin::Database#HOOKS)

Edit: I spun up a test:

Create a DB with no tables:

$ sqlite3 test.db
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In config.yml:

plugins:
    Database:
        driver: SQLite
        database: 'test.db'
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app.psgi:

#!/usr/bin/env perl
  
use strict; use warnings;

use FindBin;
use lib "$FindBin::Bin/../lib";

use TestApp;
TestApp->to_app;
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TestApp.pm:

package TestApp;
use Dancer2;
use Dancer2::Plugin::Database;

hook 'database_error' => sub {
    my $err = shift;
    warn "\nThe DB error: `$err`";
};

get '/' => sub {
    eval { database->quick_insert('no_such_table', { foo => 'bar' }) }
+;
    'Hello, world';
};

1;
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Log output:

HTTP::Server::PSGI: Accepting connections at http://0:5000/
[TestApp:4783] core @2018-09-23 09:52:38> looking for get / in /home/n
+ick/perl5/perlbrew/perls/perl-5.28.0/lib/site_perl/5.28.0/Dancer2/Cor
+e/App.pm l. 36
[TestApp:4783] core @2018-09-23 09:52:38> Entering hook core.app.befor
+e_request in (eval 305) l. 1
[TestApp:4783] debug @2018-09-23 09:52:38> Adding sqlite_unicode to DB
+I connection params to enable UTF-8 support in /home/nick/perl5/perlb
+rew/perls/perl-5.28.0/lib/site_perl/5.28.0/Dancer2/Plugin.pm l. 543
[TestApp:4783] core @2018-09-23 09:52:38> Entering hook plugin.databas
+e.database_connected in (eval 305) l. 1
[TestApp:4783] core @2018-09-23 09:52:38> Entering hook plugin.databas
+e.database_error in (eval 305) l. 1

The DB error: `DBD::SQLite::db do failed: no such table: no_such_table
+` at /home/nick/dev/TestApp/bin/../lib/TestApp.pm line 11.
DBD::SQLite::db do failed: no such table: no_such_table at /home/nick/
+perl5/perlbrew/perls/perl-5.28.0/lib/site_perl/5.28.0/Dancer/Plugin/D
+atabase/Core/Handle.pm line 310.
...
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(Note that the DB error is also printed to the log by default; you should be able to see that in your log and only need to use the hook if further processing is desired.)

Hope this helps!