Of course! Actually talking like this gives me something to do and think about other than my situation so I appreciate it a lot too.
I hope the psychiatrist can help but yes I agree that an in person visit would be much more beneficial. But, you gotta take what you can get. Are you going to be present during the session? I'm curious if you've given that some thought (if you're even able to make that decision) because I could see a case for her having a confidential talk without you but also a case for having you there supporting her. Maybe a mix of sessions?
I think the word you're looking for for mom is narcissist. Luckily, or unluckily, there are a lot of people out there who have parents like mom, so that means there's lots of spaces online that, once she's older, she'll be able to utilize and maybe cope, or at least have people to relate to and vent. One place I know of is Reddit's r/raisedbynarcissists. I don't really like to link things since I personally don't click links on forums but at least with that you can search for it yourself. If you have time it might be worthwhile to look at the top rated posts and see if there's any tips to help your daughter. Actually, if you made an account I'm sure you could post about your daughter's situation there and get some feedback by people who lived through it what they think would've helped them when they were her age.
Also, is mom the only one in the area who can watch daughter? I ask because I kind of think anyone else, grandmas or grandpas, aunts or uncles, might be better for your daughter. Of course, I know it might be tough with your work schedule, just trying to help you consider all options and why I think they're important. The reason why I ask about other options is because I imagine spending time with a mom who doesn't want you (even if mom never says that outright, her attitude, or "aura" might give that off. Kids are very sensitive to that stuff) is a lot more painful than getting almost that same attitude from someone who isn't her parent.
But, if you really have no choice, obviously she'll have to work through this. It'll be hard, but at least she does have you in her life. Things could be much worse. She could be stuck with just mom. Lots of kids go through that unfortunately.
I'm trying to decide if it'll be helpful to her bringing that up. Probaby not right now because she's pretty young and might not understand. Plus she has the accident hurdle to overcome first. But, that might be good to keep in your back pocket whenever it seems right.
That said, do be careful and mindful of your daughter! I have a recent example of my dad trying to be nice and, luckily, I was in a pretty bad place when he said it, so at the time it did make me feel better, but looking back if I wasn't in the mental place that I was it probaby could've really broken me.
So, a few weeks ago when I was instructed to go to the hospital for my situation, I was very distraught, calling my folks to let them know the news. My dad said "you know, your grandmother miscarried twins. If she hadn't I probaby wouldn't be here". I was basically rock bottom for me in that moment anyway, so knowing I wasn't the only one who had gone through this made me feel better, but the next day I reflected and thought "you know, that's probably not something you want to say to your crying, pregnant daughter who's going to the hospital" lol