I have a 10yr old D, (plus I have had two D's go through that age already), and whilst you W may be right about why your D was reluctant to commit to your invitation, I think it much more likely that your D is aware of the tension between you and your W and she is upset about upsetting your W by being too keen to do anything with you.
From previous posts you have made I have felt that your W is using the children and manipulating them which I find absolutely unforgiveable. I strongly believe that whatever happens between a H and a W, the children need to see both parents as putting the children's interests first and NOT using the children in order to score points.
I think your W needs this spelling out to her. I think she has learned very quickly that you back down when the children are around and she kicks off. She now uses that information....and the children themselves, against you. It's like she wants to belittle you in the children's eyes and that is not fair.....whatever she may feel about you. I am just not sure how to counter what she is doing as you don't want to take her on in front of the children. I need to think on this.
Saffie me 46 H 46 M in 1986 D20,D18,S16,D13 H's A 01/05 to 07/06 H recommitted to M 07/06 renewed vows 09/06 Going from strength to strength