For the current event, I think the way you tell someone that you don't want them to attend would be quite simply that: you tell them.
Barring them from the event physically brings up its own problems if they still show up.
And the fallout after telling them is going to be an inevitable mess of drama if they decide to make it one, regardless of what "the truth" is.
Ultimately you will need to decide for yourself what matters most. Dancing around the bush on it is not going to make it easier. If you choose a route like this, a clearly defined set of issues is important in explaining why, and it helps if they're more explicit than just citing a bunch of interpersonal conflicts that starts to sound more like highschool than anything else.
Ideally, in a situation where nothing has been discussed, there are no actual rules or codes, and you just don't want that one person present because of things they've done in the past, you would offer them a course of action that would allow them to attend if they can improve their behavior, assuming said behavior has not crossed certain lines of actually harming others (versus "merely" being an inconvenience to organizers?).
So how should I/we handle this problem? Of course, we don't want to
hurt Alice unnecessarily, but we also want to make very clear that we
don't want her at the next event
Personally I think you're being unrealistic. Alice is going to feel hurt if you single her out to not attend. Things have clearly gone on too far, and either Alice is aware of the issues with attending and won't, or you will be bringing things up to someone who doesn't realize they've been doing anything wrong and will be unlikely to admit that their own behavior is the issue.
It's nice that you don't want to hurt Alice, but I feel like stepping around her feelings on this is only going to make things worse. There's no nice way to tell someone you're disinviting them/barring them from attending something with otherwise open attendance, particularly when you're not instead saying "look, we need to discuss some problems which occurred the past few events, and can't have this behavior occurring again" instead of simply "don't attend."
There are ways to try to de-escalate/minimize any conflict inherent in confronting Alice about this (who is present initially, the setting, how the issues are cast), but ultimately if you have no interest in giving Alice a path to attending I think feeling like there is a way to do so that isn't going to result in some degree of definite hurt to Alice's feelings is unrealistic, because ultimately this is likely to feel unfair to Alice and like she is being singled out, which is not altogether untrue either, in how you've described the situation, even if you do have reasons for doing so.
The best way to mitigate the impact of all of this would be to address the behavior rather than the person (in which case the best approach is not to be accusing about past events in terms of responsibility, but rather to address it in terms of what can't happen at future events), but if you don't want Alice to attend the current event whatsoever, I'm not sure you have many realistic options other than being direct in simply telling Alice not to attend.
Is the problem really Alice, or is the problem the things that Alice does?
Once an event grows beyond a circle of friends, whether it's a professional conference or even simply a get-together, it goes past the point where you can actually just rely on people to "generally behave appropriately."
Can you isolate the behavior from Alice that is actually toxic versus merely frustrating?
Frustrating behavior can largely be deferred. Suggestion boxes and similar things are an easy way to push someone away. Giving them actual responsibility and then holding them to the tasks given to them can be another, if they can be trusted to do an appropriate job.
Sometimes someone who is being a busybody or similar actually does care, and they just need to be channeled into applying that caring to something productive rather than something that's creating more work for everyone else. It's also a good way to deflect any related complaints if they turn down the offer of a particular position, because you did try to give them a way to participate in how things are run. It's usually important, if you go this route, that you create a fairly constrained and specific role for them to act within, that gives them oversight but only over a specific element of the event, and where some extra attention would actually be helpful.
Toxic or even abusive behavior, particularly when it's something that can be clearly delineated in harming others or their experience at the event, is another problem entirely, and it's something that needs to be consistently addressed.
Consider a Formal Code of Conduct
If this is going to be a recurring event, and you've grown past 100 attendees, it's probably a good time to consider instituting a Code of Conduct and a related reporting structure.
People frequently mistake Codes of Conduct for being some terrible thing that impugns a community simply by being brought into existence. Instead, the ideal is that a Code of Conduct means that such behavior won't occur to begin with, and can be dealt with fairly in an organized fashion in the worst case event of something happening that does fall under it. It sets forth what you want to stand for as a community and sets an acknowledged standard for the event of expected behavior, with clearly outlined consequences for not behaving appropriately (while everyone likes to believe they are adults, not everyone chooses to act like one—or worse, they choose to abuse that expectation of appropriate behavior in order to prey on others—and the pretense that such openly acknowledged and delineated rules aren't still needed among adults is a false one in groups of strangers, even professional groups). This encourages people to come to events where they might be afraid of how they would be treated, while clearly signaling what will and won't be tolerated to others.
And finally, if Alice can't take the hint, something like a Code of Conduct sets clear guidelines for banning someone from future events who is causing problems.
Simply put, if you are organizing a repeat event hosting over a hundred people, I would highly advise looking at frameworks adopted by various conferences of similar sizes. It may seem like a bunch of bureaucracy and busy work, but once you already have events that require an organizing committee it's actually the type of thing that makes things run smoother in the long run, and can even help to stave off pockets of toxicity developing within your community and being allowed to persist far past when they should have been ousted.
It also provides a clear framework for dealing with problem behavior where the focus can be on the behavior and not on any particular individual engaging in that behavior, which helps deal with related fallout from actually addressing problems. The key here is that you have set a consistent standard, and then applied it consistently, which emphasizes that actions taken are fair and fair warning existed in regards to the consequences enacted.
The Citizen Code of Conduct and the Contributor Covenant are two good places to start when drafting a code of conduct, just off the top of my head. While it might be too late for the current event, it's something to consider having in place for the following ones. Sites which revolve around conference organization and maintaining/building things like open source communities are likely to have more good advice about drafting a strong Code of Conduct than would fit here.