In this case, I'd probably use Exception::Class objects for each possible type of failure, or else one that contains the error code. For example:

eval {
  $client->blinky_blinky;
};
if (my $err = $@) {
    if ($err->isa('NotFoundError') {
        # do something
    } else {
        $err->rethrow;
    }
}
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In general, when I want to check for a specific state, I make a method for it rather than expose internal values:

$client->response->is_ok;
$client->response->is_not_found;
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The HTTP::Response class is a good example for this kind of stuff as well.