I have written similar invites, but I hope that I have used more tact.
If I was to have the same party and request the same things my invite would look more like this:
Personally I do not see this as rude as saying there will be afternoon tea if you provide it, it is just politely asking for you to bring something if you can. By saying she is child 1's guest, I would hope that would come across as ONLY child 1 has invited you so bring something for ONLY that child if you can. I think the rudest part of the invite that Tinks has recieved is the line that says 'afternoon tea will be provided' before asking you to bring some. Bringing something in itself is not rude, it is just the wording that leaves a lot to be desired.:
Dear DD
Child 1 is turning 10 and would love you to help her celebrate as one of her guests.
She will be having a joint party with Child 2 &3.
Time
Location
Please bring a plate of food to share if you are able

