Wow, the two of you both need work. She already let you know this is her daughter's party and you are blatantly looking for a way to get around it. She already told you, "No".
If you tell your six year old this is her party too, don't you think the two young girls will talk about this and let the cat out of the bag? Personally, I think you are asking for big trouble.
If you don't like the idea of her daughter's party on your daughter's birthday, don't go. She already gave you the pass to do so. I am certain everyone knows it is your daughter's birthday and will acknowledge that without your strong arm, if you go.
My daughter's best friend's sister's birthday is on the same day as my daughter. The two girls are good friends too, but the BFF is between my daughter and the younger sister. Last year they went out of town for her birthday. This year I had not heard anything, so I told them my daughter's party would be on the day of and she would love for them to be there. In a few days, they announced her party was the next day. I took it this year and if they are just as close next year, I will give them the opportunity for first pick.
You have to work it out.
EDIT--momwithacamera did you even read my post? [yes] No she didn't give me an out [It wasn't a very nice out, but she tells you here: if you can't make it, it's okay, "i understand if you want to do your own thing and you can't make it. "] and she didn't say "no" to anything [She made it perfectly clear that we could do a family celebration at a later date. That means no, from my understanding] [No is mentioned again here: if you want to do your own]. And OMG, I'm going through a thought process, who the hell am I trying to strong arm here? [I'm also thinking of telling my daughter thats its her party too (even though its not)] I haven't done anything yet, I only found out about it a few hours ago and I'm trying to problem solve. I'm looking for help, not accusations.
My comments are not meant to insult or condemn you, but for what ever reason, you SIL has made some choices and made them quite clear. If you interfere, I think you are going to risk upsetting the closeness of the family. I'm not saying what she has done is right, but I can see how this whole thing could very easily go down. Maybe I am wrong, maybe she will change her mind and say bring your cake over and lets include her, but if that were the case, I think the two of you would have had a different conversation. Based on what you state, this wasn't an oversight.
Perhaps she chose that weekend because it is not as close to Christmas. I can't tell you why she is not willing to include your little one.
There are many of who are cheated out of their special day, due to the day we were born. I was born on Mother's Day. I robbed my mother many years of a Mother's Day and now it is my turn to pay it back. I shop, clean, and cook on Mother's Day...my birthday. Everyone else in the family gets a special birthday dinner but me. If I would like, I could plan my birthday to be celebrated on the next weekend or the weekend before.
I am trying to tell you not to get so worked up over a specific day. She is not leaving you much choice. Next year, [politely] plan your party and pass out your invitations. If you so choose, include her daughter.