Hi,
I’m afraid there is no such option, nor an easy way to add it programmatically, as re-Captchas have to be rendered on both client and server side. I usually recommend a custom code where I can, but this is far more complicated.
Having a re-captcha within the search form is very unusual and it will not achieve anything, as the requests are secure on their own. Browsers use a no cross-site policy, so fetch/xhr requests can’t be made outside of the website scope.
If you want to have the website secured, then I rather recommend using Cloudflare CDN with the highest security settings. That will automatically trigger a captcha to the visitors which are suspicios or flagged.