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No errors with the HTTPS. Checked Ahrefs: very few direct anchors. Most anchors are on their real name from big publications and other websites citing the company... These websites always write topic-relevant content (and include the keyword close to the anchor). I'm desperate with this client

Hahaha. I think you might have one of the few instances that I recommend something I generally don't recommend, which is a hreflang across domains. Now this is going to get complicated, so let me know if I miss something in my explanation. First, I normally would recommend that the client just use the English translation of the main site if the content won't change at all. But you can't due to the trademarked name. Second, it sounds like your client never intends on changing the content on the new site for the US audience and doesn't need to. I assume this means that there is no change in products/services and no reason for people to see different content between the France specific English version and this other English version. If there is any change in content, like imagery, messaging, adding, modifying or removing products or services, please let me know. That changes the answer. If there is no actual changes in content, none at all, and no reason for them, you'll want to use a cross domain hreflang. France-French bbb.com France-English bbb.com/en  English (separate brand) aaa.com I would use hreflang to show that bbb.com is translated to en-fr at bbb.com/en and en-us at aaa.com. That's how to deal with that situation, but if their site is a .com and there is some way to just offer their content in English, that will get results faster and be easier to maintain. With the setup you are considering, you would still need to work on promoting the new domain and that will take time. I hope this all made sense.