The following program passes its tests for me, side-stepping the source code encoding:

    use utf8;
    use 5.16.0;
    use Test::More;
    use Test::utf8;
    use JSON::XS;

    BEGIN {
        # damn it
        my $builder = Test::Builder->new;
        foreach (qw/output failure_output todo_output/) {
            binmode $builder->$_, ':encoding(UTF-8)';
        }
    }

    foreach my $string ( 'Deliver \N{LEFT-POINTING DOUBLE ANGLE QUOTAT
+ION MARK}French Bread\N{RIGHT-POINTING DOUBLE ANGLE QUOTATION MARK}',
+ '日本国' ) {
        my $hashref = { value => $string };
        is_sane_utf8 $string, "String: $string";
        my $json = encode_json($hashref);
        is_sane_utf8 $json, "JSON: $json";
        say STDERR $json;
    }
    diag ord('»');

    done_testing;
[download]

On the Windows command line, I get ord('»') also as 187 for perl -wle "print ord('»')"... I'm not sure what to make of this, but I feel confirmed in my intention to have no characters above code 127 in my source code...

Thanks for introducing me to Test::utf8 - this feels useful when diagnosing mojibake issues.