What is the closest result relating perfect shuffle closure of primitive words?
I am not aware of any result of this type. And as your example shows, the set Q of all primitive words is not closed under this operation. 

You can find subsets of Q which will result only in primitive words. For example a a+ b b+. The perfect shuffles will be from a a+ (ba)+ b b+ and all of these are primitive again (though not from the original set). 

The results on square-free shuffle seem to be not about the perfect shuffle, but the general case. Then they say that at least one square-free word is in the shuffle. That should be easy to show for primtive words. In your example, all resulting words are primitve except the one from the perfect shuffle.