I would use invisible recaptcha. Then on your button use a tag like " formname='yourformname' " to specify which form is to be submitted and hide a submit form input.
The advantage of this is it allows for you to keep the html5 form validation intact, one recaptcha, but multiple button interfaces. Just capture the "captcha" input value for the token key generated by recaptcha.
<script src="https://www.google.com/recaptcha/api.js" async defer ></script>
<div class="g-recaptcha" data-sitekey="yours" data-callback="onSubmit" data-size="invisible"></div>
<script>
var formanme = ''
$('button').on('click', function () { formname = '#'+$(this).attr('formname');
if ( $(formname)[0].checkValidity() == true) { grecaptcha.execute(); }
else { $(formname).find('input[type="submit"]').click() }
});
var onSubmit = function(token) {
$(formname).append("<input type='hidden' name='captcha' value='"+token+"' />");
$(formname).find('input[type="submit"]').click()
};
</script>
I find this FAR simpler and easier to manage.