The main goal of my response is to challenge you on some of what you said in your question, which hopefully will allow you to have a calmer and more productive conversation with your daughter.
Since I feel I'm quite judgemental myself in my answer, I'd like to add here that all I have to go on is your question. Those are the impressions your post have left me with, and even if you think my assumptions are wrong, I encourage you to do some introspection to see if there is any truth to what I'm saying here.
Aggressiveness rarely comes from nowhere
You say she becomes aggressive when you discuss any real issue with her, but it wouldn't surprise me that she's only responding to your own "aggressivity". The words you've chosen and the perspective through which you see your daughter's action are quite strong. You're "immensely disappointed", you're a "moral compass", you have a "duty to speak up for the family". You're using a lot of words which elicit strong emotions (and quite a bit of judgement). And then there is this :
I feel like by wearing this coat all the time, she is going to great lengths to reject not only the way she was raised, but in essence, me.
Maybe this is true, maybe she has chosen a coat to spite you (in that case, I would first ask myself why my daughter feels the need to spite me like that, those things seldom happen without reason). But the likeliest truth is that this has nothing to do with you. Nothing. She didn't think about you at all buying that coat, just how pretty it is. We as people tend to take things very personally, while often the person "offending" or "attacking" us didn't spare us a thought at all. It is very possible that the explanation your daughter gave you (feeling bad it has fur but loves the coat anyway) is all there is to it.
This choice is in conflict with what I believe are our family values.
Even if this is actually something that your whole family feel strongly about, by using that kind of language you are alienating her from her family. Worse, you're making her an enemy, someone other. Don't speak for the whole family, and just express your own opinions and feelings. The fact that it's an important issue for you is enough.
Understand this from the perspective of your daughter : she does something with probably little thought, and you come along and accuse her of abandoning family values and disappointing and rejecting you and the whole family. If the roles were reversed you'd probably feel attacked yourself. Even if you don't use words like that when talking to her, the fact that you have this opinion of her will color your tone, your facial expressions and what you say. She'll know you're upset, she'll know you're on the warpath. She'll respond in kind.
When I want to talk to someone about something that they have done that upsets me, I like to reframe things in my head. It's rare that someone actually wants to hurt you or do things just to spite you. And no one likes to be accused of such things. So instead I take a minute to reframe the issue in my head from "They wronged me !" to "They've done something that ended up hurting me, I'd like to have a conversation to know why they did what they did and express how their actions have affected me". This allows me to have a calmer conversation and have an open mind during that talk. People often sense that and respond in kind, meaning they feel safe to explain their side and even apologize (instead of getting defensive). No one likes to be lectured, even less about something that wasn't their intention.
Moral high ground
A lot of the words you choose gives the impression that you are morally right, and your daughter is morally wrong. As long as you give of that vibe, your chances of having a meaningful conversation with your daughter aren't high. I'm a vegetarian transitioning to veganism. Just saying this, many people feel judged, because it is, for many, a moral issue. Also, many people react aggressively because they have had repeated experiences of higher-than-though extreme vegans or animal activists. To have a conversation (not a lecture) with people, I have to put extra effort to make them comfortable, to know that I'm not in fact judging them, and that I'm not here to harass them about their choice. That if they have questions, I won't mind educating them on the subject. I'm not here to tell them how bad and wrong they are.
...although we are a meat-eating family, I tend to buy only organic or ethically-raised meat and try to reduce plastic waste.
She mentioned when she bought it that it kind of bothered her that an animal was killed for it, but she went ahead in spite of that.
What I feel when I read your statement about still eating meat must be quite close to what you feel towards your daughter's choice. What you find lacking in your daughter's explanations, I find lacking in yours. However, I'm aware that this is a personal choice. I don't do it to be better than meat-eaters, and people who eat meat don't do it to spite me.
Your daughter probably thinks you believe to be better her than her. That she is less for having that fur coat. The words you use tells her you're not interested in discussing this with her, you just want her to hear what you have to say, and agree with her. In her place, I would be very frustrated too and unwilling to talk with you. If this is how your "talks" on real issues go, she's maybe not even willing anymore to have discussions with you if she feels her opinions don't matter and are discarded if they don't align with yours. This all means that to be able to have conversations with your daughter, where she listens to your arguments (and you listen to hers, and respect her), it will probably take time. I strongly encourage you to self-reflect and really think how you talk and treat your daughter, and if you have played any role in the fact that you two can't seem to have civil conversation about sensitive subjects.
Know what to ask
How can I express my disappointment about the coat to her, without causing a dispute?
Expressing your disappointment about the coat will cause a dispute. Or her shutting down. I can almost guarantee it. If your only goal here is to make her aware you disapprove and she's making a terrible choice, I'd ask you why. Why do you ask us how best to berate your daughter, instead of how to have a conversation with her about her choices and the impact it has ? Since your conversations about other sensitive issues has gone bad in the past, I think your best bet is to make her think about her choice and let her come to her own conclusions, instead of just telling her what you think. Something like :
"I'd like to ask you about something. I've done some research on Canada Goose, and I was quite surprised when I learned about their fur farming practices. Are you aware of them ?"
Then let her respond.
You also had a second goal in your question, which I think is more achievable (and reasonable).
...it's about enjoying her company and my time with her, and not being afraid of blow-out meltdowns.
There will be only blow-out meltdowns if you press the issue every time you see her, I don't think those happen without provocation. And I don't think you'll be able to force her or expect of her to not wear the coat, so I would abandon that goal. What you can do is express how the sight of that coat makes you feel, and ask, as a favor, that she tries to not wear it when she knows she'll see you (you can't force her though). You can do that after she tells you why she finds fur coats ok :
"I just always thought animal welfare was an important subject for you too. As you know, it's an issue I'm very sensitive about, and seeing that fur coat makes me deeply uncomfortable. I would very much appreciate if you try not to wear it when we're supposed to see each other. If that's not possible, I at least don't want it in my home, thank you for understanding."
