Anaïs - I have always loved. I used to think it'd be out of the question because of pronunciation difficulties in the UK, but I wonder if I've been hasty. I think many won't know how to say it at first (Many seem to say 'Anay'), but once corrected An-eye-ees seems achievable. It's still not the cleaner An-a-ees pronunciation that you'd get in France though. What do people think?
At the risk of getting flamed for being classist, it depends where you live in the UK. If you live somewhere with a lot of ethnic diversity and/or a lot of CSP+ inhabitants, it will be fine. As you say, not a perfect match in pronunciation, but an approximation that's pretty close. If you live somewhere with more homogeneity of names, then anything out of the ordinary risks being mangled.
That said, I agree that most people would be able to remember the approximative pronunciation once they were told. It's not a difficult name for an English speaker: no silent letters, no letters that make a sound you wouldn't expect, no sounds that don't exist in English (techincally that last point is slightly incorrect, but you've said that you're ok with that).
The other, small, issue, is that English keyboards still aren't set up to use diacritical marks very easily (you can install your own shortcuts or use the alt keycodes, but it's not as straightforward as a French keyboard). And some electronic databases and the like don't accept them. Nowadays a lot is done by phone rather than computer, where diacritics are much easier, and most companies are getting better. But there are still enough difficulties that I would think twice before giving a UK baby a name with anything other than the standard 26 letters (I speak as the holder of such a name). For example, you can't get a UK passport with a diacritic, you would just have to write Anais on the passport, which is annoying (at least to me). And it's a bit hit and miss whether they would spell it properly in primary school. The other things are smaller fry, like sports.
I wouldn't rule out Anaïs. If you like it, it's perfectly manageable. Just be aware of the potential pitfalls.