Thanks for sharing your new site. In digging into this, I’m wondering if this snippet from Google’s reCaptcha docs might be helpful:
I see that you have something like this in your <head> tag:
<script src="https://www.google.com/recaptcha/api.js" async defer></script> so that’s good to go 
In the body of your form, could you try adding this:
<div class="g-recaptcha" data-sitekey="your_site_key"></div> <br/>
And delete 'g-recaptcha-response': captcha from your POST body in your JS? Alternatively, you could try changing 'g-recaptcha-response': captcha in the POST to 'g-recaptcha': your_site_key.
Let me know how that goes- fingers crossed!