I think that if we do it that way, we risk alienating people who are a little slower at posting. What if it was less about a specific posting order and more about groupings?
For instance, right now, the established posting order would be:
1. Nevermind
2. TWCross
3. Thought Manifest
4. Whirlwind
5. Shadowcatcher
but instead we could see it more like everyone has posted once, so everyone is at liberty to post again in any order. We just can't start the third round of posting until everyone else has made their posts. Does that make sense? I can't really try a proper visual, but I'll try to reason it out better.
Let's say that Whirlwind wanted to post next (in the second round of posting) even though Nevermind, TWCross, and I haven't posted yet. That could be okay. Basically, she could post, but she wouldn't be able to post again until the rest of us have caught up. Once the four remaining players have posted, she can post again or wait until another person posted. Or... to give you an idea using yourself (shadowcatcher), once someone else has started the second round of posting, you could make your post at any point instead of having to wait until the very end again.
We'll need Nevermind's approval, but I think that way might be better for those of us with busier lives. If each player can post after just two posts, it's possible that one or more of us could be left in the dust while two or three characters were able to interact without the rest of the characters putting forward their reactions, too. I hope that all made sense. If not, I'll try explaining it better.