Invisible reCaptcha for WordPress.

можете попробовать display: none
но тогда вам придется разместить ссылки на гугл:

Can I just remove the badge? (legal)

Yes, according to Google’s FAQ on reCAPTCHA you can hide the badge. But you have to put a note in the “user flow”.

    You are allowed to hide the badge as long as you include the reCAPTCHA branding visibly in the user flow.
    https://developers.google.com/recaptcha/docs/faq#id-like-to-hide-the-recaptcha-badge-what-is-allowed

Google suggests this text for this purpose:

    This site is protected by reCAPTCHA and the Google
        <a href="https://policies.google.com/privacy">Privacy Policy</a> and
        <a href="https://policies.google.com/terms">Terms of Service</a> apply.

коды

.grecaptcha-badge {
    display: none;
}

альтернативно:

.inv-recaptcha-holder {
  visibility: hidden;
  opacity: 0;
  transition: linear opacity 1s;
}

.inv-recaptcha-holder.show {
  visibility: visible;
  opacity: 1;
  transition: linear opacity 1s;
}

$(document).ready(function () {
  $('form input, form textarea').on( 'focus', function() {
    $('.inv-recaptcha-holder').addClass( 'show' );
  });
});