こういう感じでMaltsのto_appを上書きすればいい気がする。

Web.pm

package MyApp::Web;
use 5.10.1;
use strict;
use warnings;
use parent qw/Malts Malts::Web/;
use Log::Minimal;

# ...略

sub to_app {
    my ($self) = @_;
    return sub {
        my ($env) = @_;
        local $Log::Minimal::DIE = \&_throw;

        my $res = eval { $self->SUPER::to_app->($env) };
        return _catch_error($@) if $@;
        return $res;
    };
}

sub _throw {
    my ($time, $type, $message, $trace, $raw_message) = @_;

    die {
        code => 500,
        message => 'Unknown Error!',
        time => $time,
        type => $type,
        trace   => $trace,
        (ref $raw_message ne 'HASH'
            ? (message => $raw_message)
            : %$raw_message),
    };
}

sub _catch_error {
    my ($e) = @_;
    if (ref $e ne 'HASH') {
        $e = {
            code => 500,
            message => $e,
        };
    }

    my $c = Malts->context;
    my $res = ...; # レスポンスオブジェクトを作る。
    return $res->finalize;
}