There is a lot to unpack here and I don't know how well my tired brain will do with this, but here'goes.
I think you should definitely (as gently as possible) let Anne know that your feelings are changing. If there is anything at all that is going well, let her know some of that too, but also let her know what your new needs are. If you need to step all the way back from having a relationship with her right now, let her know that.
You feel that you are completely not okay with long distance relationships. That's fine. And if you want to let her know that your feeling may change even further if she goes, that;s honest and fine too. But please, for the sake of everyone, do not try to coerce her into staying. That would be mean and unfair. Not to mention that it could have many lasting and negative consequences whether she decided to go anyway or was talked out of going.
I'm glad that you and HMA have an outlet like this for when communication becomes difficult. In my view, all of these kinds of concerns are best addressed as soon as you can be ready to communicate them. Sometimes it's VERY difficult to be ready to communicate about difficult things, but sitting on them can make them blow up like a baloon sometimes, and the better you get at discussing them openly and honestly (and with all involved parties if possible) the easier things become in the long run.
Sometimes emotions that totally don't seem like they can or should go together still happen at the same time. You expect to feel happy, and actually do feel happy, but feel overwhelmed by grief or fear at the same time. Give yourself permission to take the space you need to sort through things. I feel like it's better to communicate soon rather than later, but try to figure out what forms of processing work for you, what kinds of things help you get ready to talk (or scream, or have a pillow fight, or whatever's appropriate to the situation). I've known people who've needed to go to a certain place or listen to certain music or talk to a certain person or do various things that had some significance for them that helped them sort out their thoughts.
Now, this part is kind of important. At some point in the past you entered into a relationship that involved three people. So there are now three people and multiple relationships to consider. If you wish that things could go back to normal/easier/etc. that's the emotions that you are feeling and how you feel is how you feel. But how you feel has a funny way of not affecting reality at all sometimes. And, more importantly, the only person you can really do the feeling for is yourself. You can't make HMA feel his feelings for Anne are going to change if she goes away for a time. You can't make Anne feel any particular kind of way. You can only do your best to move forward from where you are (and not where you were or where you wish you had gone).
Like is extremely complicated all on it's own. And finding yourself in a lifestyle that complicates it even more can be enough to make you want to scream. (and screaming is okay.) But I have found in this lifestyle that there are even more days where it feels like a huge benefit to have more than one love that you can go to and days where this lifestyle can seem so simple and uncomplicated that you just want to sing.
(and singing is okay.)