My question involves a child custody case from the State of: Tennessee

Hello,
I was wanting to get some advice about a current situation. Me and my ex have a 6 year old daughter. I have full custody, he has visitation, I get child support etc...
Well, we had her when we were teenagers and he didnt see her much, but maintained an ok relationship with her. When he couldnt afford insurance or child support he joined the Military. He is stationed in California. In the past three years we both have gotten married to other people. Our daughter has lived with me and my husband the last 3 years. Her dad did not see her for 2 years. He said it was for financial reasons, couldnt come back here to see her, and not having enough leave, etc... Whatever his excuse was.
He came into town on Monday and asked to see her for fall break. I agreed to let her stay with him and his wife at her parents house for a few nights. Well they have done fun stuff, she had a great time, and the mental picture she had of her daddy, apparently is coming true for her. He has admitted to me that he was not the best father in the past but he is older now and more mature and wishes to spend more time with her, etc..
Well he is going back to Cali in a few days and she wants to go with him, because they havent had much time with him.
I talked this over with my mother about getting a temporary order stating that she can be with him until the school year ends, and me suspending child support until she comes back.
He really wants a chance to be a good father to her and most of all I want my child to be happy.

So my question being, if I get a temporary order for her to spend the rest of the school year with him, and agree to suspend child support, can he then sue me for child support and could it hurt me later, if he decides to sue me for custody?

I dont want to lose my daughter, she is my pride and joy, but I also dont wanna deprive her or him of having more time together.
Advice?