If it were me, if your birds were mine, I Wouldent, ever let them out at the same time. Not after I seen that. I usually trust my gut and my gut is saying no, no. If you want your birds to be healthy and happy, then no, not even letting her sit on top of his cage cause he might try to take off a few toes or something.
And I think its strange, because arnt birds flock animals? Shouldent two birds try and be buddies, rather then attack eachother? Ive heard of birds getting upset with smaller birds, like budiges for picking at them, but I never heard of parrots closer to the same size not being friends.
Ive watched it close to 15 times now, and (What I feel inside says) it looks like the dance is for YOU. Hes made a show of trying to get out, trying to find a opening between the bars into a dance, and you speak to him showing approvial and acceptance, so he makes a display, but the main thing I see is him trying to find a way out, checking the bars over and over again.
"theres a gap here, no too small, theres a space here, no too small, damnit, I cant get out, mom open the dooooorrr Pleaseeeee"
The more he dances, (Cause it IS vary cute if you look past the pausing and fluffing, every time he watches you, or your gray move.)
Again, I am not an expert, I have never own a bird on my own before, and like a lot of other people Cherry, larry, the Moluccan cockatoo, and all the budgies Ive house setted, dont count. but I have worked with animals all my life, I was 5 years old when I got my first flock of 12 chickens and tended my grandmothers flock of 10 geese every year after that. Ive learned to watch.. and it has worked for me, the last few years it has become vary handy.
Id beware. JUST please be careful, Ive seen small birds kill other birds before.. its not a pretty sight.
. But he sure is a very beautiful bird 
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