As a teacher and a parent I understand completely what you are going through. Third grade is the hardest elementary grade. Students are all of a sudden expected to do so much! I have seen many students repeat the third grade and excell years later. If she knows that the work won't be as hard because it is something she has seen before, and if she knows that she is not ready for 4th grade, it might be a bit easier. I would also remind her that students that are a little older than the others have an advantage academically. If she is social and has a good self esteem, she will flourish.
Good luck and God Bless.
I think that every solution you have considered thus far is exactly the right thing, assuming that you've also decided on her repeating the grade. Also, I agree that her beginning to mimic a class clown could be directly linked with her not knowing what to do. It's actually an advantage that she is younger than her other peers so that this isnt necessarily contributing to putting her too far behind by repeating. Id be against it if it were putting my child in a situation to be graduating high school at 19 or 20, see?
When you do tell her about her repeating the grade, simplify it in terms such as "We've decided that you repeat this grade so that we know for sure that you are covering and understanding all of the information that you need." Dont go in depth with her because if you were to start to point out what she didnt do well in, she could develop some insecurity and anxiety issues.
As for your feelings, approach this upcoming school year enthusiastically, readily, and wholeheartedly. Remember, if she sees or hears your concerns that you are sad or disappointed about this, she will be too. If you consider it as labeling her as a failure, that's how she'll feel about it as well.
As far as changing schools, I dont feel its necessary considering the grade level. I have childhood friends that repeated early grades and they honestly can hardly remember what grade it was that was repeated. Its actually that older grades (5, 6 and so on) where the children and peers may take notice.