This is one of those cases where inheritance isn't necessarily the right way to do things, even though it seems ideally suited. What you really want is delegation, specifically you want to decorate the ExceptionThrowingModule with your own module. So, something like so:

package MyModule;

use OtherModule;

sub new {
    my $class = shift;
    my $obj = OtherModule->new( @_ );
    return bless \$obj, $class;
}

sub AUTOLOAD {
    my $self = shift;
    my ($method) = (our $AUTOLOAD =~ /([^:]+)$/;
    return eval { ${$self}->$method( @_ ); }
}

# These are in UNIVERSAL, so need to be overridden because
# AUTOLOAD won't catch them.
sub can {
    my $self = shift;
    return eval { ${$self}->can( @_ ); }
}

sub isa {
    my $self = shift;
    return eval { ${$self}->isa( @_ ); }
}

1;
__END__
[download]

There's more you'd have to do in order to make sure you play properly with overload and others. But, that's the gist of it. And, the runtime cost is quite minimal, probably on the order of 1-2%. And, you're very loosely couple with OtherModule, meaning that as OtherModule changes, you don't have to change with it.