Please don't do this. It is absolutely none of your business as to whether she's discussed her daughter's transition into school with the teacher.
Totally agree with this.
You can ask how she is getting on. How she's finding the longer day. You can share your daughter's own experiences (generally, not wrt Rose) and if she wants to say something she will. Starting school is a tricky time for all children, not just the ones who express it by hitting their friends. Rose doesn't deserve any more special treatment than your daughter and it might well benefit her to be able to encourage other friendships.
Also wrt the party, I personally wouldn't invite Rose. Nursery friendships are not set for life. I think it's fine to say "She's only invited 5/6 children from the class and Rose wasn't one she chose [because it is a bit cheeky to presume]. Do you fancy a trip to the park on Saturday?" or whatever.
This all feels like a big deal at the moment, but you'll realise very quickly that it happens all the time. I've not invited someone who thought they would be, mine haven't been invited to friends' parties. It's just life.
Rose isn't going to be traumatised; you're not going to be a social outcast; and if Rose's mum gets upset about it, then she's going to have to learn to toughen up pretty quickly.
No one has the right to be invited to another person's celebration.
It's fine.
FWIW... When my daughter started school, she went with one other child from her nursery. They are now friends, but Reception and KS1 were a nightmare. The other girl was unkind to my daughter. It was never physical but there was constant clinginess, competition, comparison and put downs from her that found my daughter lacking and confused and irritated the hell out of her.
Of course I explained that this girl might be finding it tricky moving to school and establishing her own place etc but, ultimately, I told my daughter that she had to be kind, but didn't have to be friends/play with someone who was upsetting her.
Because she didn't.
I had a mother who didn't ever listen to me as a child. Who never prioritised me over other people no matter how badly I was treated. This isn't the thread for it, but it taught me a very important lesson in life; that I, and my feelings, didn't matter.
Yes, this is only a birthday party, but it's also an opportunity for the OP to show her daughter that she has her back, that her feelings are valid and that, sometimes, you have to put yourself first. And if you can't do that at your own birthday party when you are 4, when can you?