Using Recaptcha shows error and breaks templates

In the last month or two we have been hit with a gazillion bot requests all from AWS EC2 instances (600K+ so far). Adding recaptcha seems like an attractive solution, but we can’t seem to get it to work.

The plugin is installed, activated and set to v3, the key and secret have been added from Google for v3.

The link to the site is currently showing the page without the recaptcha enabled, as enabling the plugin breaks the downloads and breaks our child theme template for downloads.

When we click on any button to download, we see this error in the console:

recaptcha__en.js:465 Uncaught Error: No reCAPTCHA clients exist.
    at Array.<anonymous> (recaptcha__en.js:465:17)
    at recaptcha__en.js:450:466
    at 3db6da2fa2e79ab22b8c340e4beadaa9.js?ver=6ca05:93:1259
(anonymous) @ recaptcha__en.js:465
(anonymous) @ recaptcha__en.js:450
(anonymous) @ 3db6da2fa2e79ab22b8c340e4beadaa9.js?ver=6ca05:93
setTimeout (async)
(anonymous) @ recaptcha__en.js:361
(anonymous) @ recaptcha__en.js:398
(anonymous) @ 3db6da2fa2e79ab22b8c340e4beadaa9.js?ver=6ca05:93
dispatch @ jquery.min.js?ver=3.6.4:2
y.handle @ jquery.min.js?ver=3.6.4:2

The template and and error above are only present if I enable Recaptcha Global checkbox. Disabling, everything goes back to normal, but there is no captcha. Presumably because we have not added the recaptcha tag:

[dlm_captcha id=5]

With global enabled, this is what our page looks like:

We have tried to set the keys a second time without any change. Any idea what might we might be doing wrong?

The page I need help with: [log in to see the link]