There's no requirement that you use any particular version of the Chaos Campaign for Alpha Strike.
I'd suggest using the new free Chaos Campaign: Succession Wars. (The Alpha Strike Commander's Edition does not have the Chaos Campaign rules). But if you want to continue using the Alpha Strike campaign rules from the 2nd printing Alpha Strike rulebook, that's up to you.
Campaign Operations is not intended as an Alpha Strike rulebook. There are some sections that could be useful for Alpha Strike, but it certainly was intended to be required for Alpha Strike. If you happen to already have Campaign Operations, you are welcome to use it's version of the Chaos Campaign if you'd like.
