You're right to be concerned with regard to your daughter's emotional well-being. She is in for a very rough life (it doesn't end when she leaves home) if something doesn't change. In 1986, Ney et al published a seminal study showing that of 5 kinds of abuse (physical, verbal, sexual, physical neglect and emotional neglect), verbally abused children were most negatively affected, with low self-esteem, self-directed anger, aggressive behavior and pessimism about their futures.
You're an adult, and you can handle your wife's unpredictable (or predictable) mood swings. One reason is that as an adult, you know how to keep yourself safe. You are in a situation of being equally as (or maybe even more) powerful as your wife. You have power over your physical self. You can probably also remove yourself physically from an alarming situation.
Your daughter, however, is completely helpless, and depends on her parents to keep her safe in a world that has more than enough challenges to deal with. To allow her to be subjected to a volatile and frightening parent will do a considerable amount of psychological damage.
Imagine, if you can, how you would feel about your wife's rages if you were in an accident and were consequently paraplegic with a limited ability to communicate. You are with her all day (no going off to work) and pretty much totally dependent on her. You have no way to help your wife calm down or to negotiate the situation. She's screaming down at you in your wheelchair, and you can't just roll yourself away. There's little you can do except endure her rage each time and hope she doesn't throw something at you or hit you this time. If you turn your back to her, will she get even angrier? If you can imagine that, you might have a window into your daughter's situation. Young children of emotionally labile parents are sometimes afraid their parent will kill them. Furthermore, your daughter has no recourse except to hope for your protection during these events, which are traumatic regardless of at whom the rage is directed.
Add to that the fact that young children think the world revolves around them. Your daughter will think it's her fault that mommy is so angry. She will grow up thinking she is a bad person who deserves to be emotionally abused.
You say that you don't want to leave your wife, and I want to respect that. But know that to stay, you are acting as if her raging is acceptable behavior (in action, not words).
You can't protect your daughter while her mother is a primary caretaker. As you said,
I confront my wife about it and she admits it's wrong and promises not to do it again - but she does. I genuinely don't think she has any control over it.
You may think she's unable to control it, she might think she's unable to control it, but the truth is, she has no valid reason - none whatsoever, really - to refuse to seek help for her mood disorder.
You don't have many options if you want to stay in this relationship. Some of them are
- start therapy yourself to learn why your wife behaves the way she does and how you can set healthy boundaries‡
- to convince her to go for professional counseling and to stay in counseling as long as this is a problem
- remove your daughter from this situation (can she live with her grandmother?) until your wife learns to control herself permanently
Is the latter option legal? It is if your daughter is being abused (verbal abuse counts.) Start gathering proof with your smart phone, and store the evidence where she has no access to it. Likely she will rage at this as well, but it seems to me you need to start doing something radically different if you want her to get counseling.
What you can do for your daughter is limited, but every bit helps: love her, validate her, give her an emotional vocabulary so she can express herself ('scared/sad/mad/happy/etc.), make sure she knows that your wife's anger is not her fault, and protect her from your wife's rages. She needs you.
‡Your wife blame-shifts: when she's angry, she blames someone or something else for behavior she engages in. It is likely that initially when you stand up to her, she will become angrier and will blame you. You need to be prepared (this is where your therapy will help) to handle this. It is likely to get worse before it gets better.
There's more literature here than you can shake a stick at.
Childhood Adversities, Interpersonal Difficulties, and Risk for Suicide Attempts During Late Adolescence and Early Adulthood
Sticks, Stones, and Hurtful Words: Relative Effects
of Various Forms of Childhood Maltreatment
Preliminary Evidence for White Matter Tract Abnormalities in Young Adults Exposed to Parental Verbal Abuse
Parental verbal abuse and the mediating role of self-criticism in adult internalizing disorders