Make compatible with Flatsome popup registration form.

Can you provide that plugin url?

It is a theme: https://themeforest.net/item/flatsome-multipurpose-responsive-woocommerce-theme/5484319

Here a demo: https://flatsome3.uxthemes.com/

Take a look a this capture: https://nimb.ws/oMIwLs

Let me know if you need admin and FTP access to my site.

Sorry, there is no plan to support any theme/plugin which is not hosted in wordpress org repository.

But it is very easy to implement this in any plugin/theme if you have a little knowledge in php. Please see https://www.shamimsplugins.com/docs/advanced-nocaptcha-recaptcha/getting-started-advanced-nocaptcha-recaptcha/implement-in-custom-form/. If you do not have any php knowledge i believe you will need to hire any developer.