How do I invite an expert to contribute to a review article I'm writing?
I suggest that you contact the other person and ask for their advice on your ideas, but not for collaboration, initially. See where that goes. It allows the person to make a small commitment without feeling any pressure for a larger one. 

Your idea for a review or summary article might be a bit unusual, but in many fields these are occasionally needed as the field splits but the different pursuits have some common but unnoticed features. Partly for this reason, I don't think that the other party would think about going into competition with you on the idea, and might want to participate. The advice would be valuable in any case. 

If the collaboration starts naturally then you can write a joint paper, perhaps after you complete your degree so that your advisor has less sway. Write and explain your ideas, expanded from what you write here, send a draft, and ask for any feedback that he/she might give.

The other person might also be able to suggest others with whom you might profitably share your ideas. 



I mention the competition issue only because you don't really know the person. Most everyone behaves ethically, but not all do. So in some situations it is necessary to be a bit cagey when you are a newcomer to academia. I don't really see it as an issue in this case, however, as mentioned above.