I think there are some fantastic responses to this thread already with sage advice from parents who have a few years experience of this type of scenario under their belts.
I can remember in my daughter's reception year it was all about having whole class parties in halls, because everyone was making new friendships and getting to know 20 odd other children on top of the 10 they'd been at pre school with and already had friendships with (or not). Parents seemed keen to include everyone and in hindsight it was a good idea, as the year went on children with birthdays over the next 9 months then invited the child to their party, and it all helped with being included with lots of other children in the class and different friendships developing.
I'd say that most of that first year was spent making new friends as well as continuing existing friendships or drifting away from them, social groups were pretty firmly established by the summer term and now in year 2 they are still pretty much exactly the same. It's a really important year for establishing friendships at school, and your daughter inviting some new friends is great, but if they're replacing one little girl from an existing pre school friendship group after 7 weeks at school at the party then its not so great. I know my daughter would be devastated in Rose's position, and I'd be seeing that friendship with the other child as over as well as any friendship I had with her mum. In my experience the mums can also be just as upset over party invitations, if not more so than the children!
I help out in classes at school and at this age there is huge status and excitement that builds when a child has a party. Quite often they will use it as a bargaining tool telling each other they will / won't be invited to their party if they do something they do / don't like, some of them do this even months before their birthday! It really does have a big impact on kids at this age, I've seen the devastation and tears from the children not invited to a party when the whole class parties stopped in year 1 and people started having smaller gatherings. Personally without knowing the details from the teacher, I couldn't do that to a 4 year old that's been part of a friendship group for 2 years til now, not unless I knew from the teacher she was definitely being persistently unkind to and hurting my child, and even then, call me a mug but I'd be trying to employ understanding and kindness as an initial tactic to deal with things.
Things happening at school should always be dealt with first by the school and ours repeatedly tell parents please come to us if you have any concerns. I would have spoken to the teacher to establish the facts before now, not relying on reports from a group of 4 year olds who we all know can misconstrue and exaggerate things. Please speak to the teacher first. I'm also wondering if the reason the mum text you like that is because Rose has mentioned she isn't invited / wanted and she's worried. Or she's just genuinely assuming Rose will be part of your daughter's birthday within their existing friendship group like she has been over the last 2 years, which let's face it is half their little lives and a pretty significant period to us as well as to them at that age. I'd reply saying you're still deciding what to do and then I'd get the facts from the teacher next week.