After thinking it over, I decided to create a small mu-plugin that automatically redirects people who are not logged in and try to access a restricted page to the login page.
I should point out that in the settings for this plugin, the “BuddyPress Register” checkbox is checked. The captcha is also visible on all of my forms (contact, WordPress login, etc.).
I am looking for a way to add the captcha specifically to the “Members-only area” page.