louvers or screen on 2nd floor chase

You may want to get the install inspected FIRST, then cut in your louvers.
I believe your Class A needs to be COMPLETELY enclosed to meet code.
Once the install has passed you can do what you want.
I would cut a hole at the top & the bottom & install louvers in both.
That wayyou can get a little convection flow going...

Good idea. Put a small (6x10) return air grille at the bottom and top of the chase.
http://www.lowes.com/pd_121908-45311-AB3BGWH106_1z0uaty__?productId=50275259&pl=1

(edit: Daksy's suggestion is probably a good one. Our local inspector had no issues with this but you never know. )

One local installation I helped with made a circular chase out of perforated stainless steel that enforced the 2" clearance requirement. It looked very sharp and fit with the contemporary decor. The inspector was curious but passed it with the understanding that the intent of the chase is to keep anything combustible at least 2" away from the chimney pipe including human hands. I don't see how a vent grille would violate the code intent, but some inspectors can be real sticklers.

Chimney pipe is insulated. The cooling of the flue gases in the pipe is minimal at ambient room temp.