You can carbonate it at whatever temperature you would like. My thought process is like this:
Beer comes out of fermenter and goes into keg.
I want to drink beer, like, yesterday.
If I carbonate now, I can "forget about it" in the fridge while it comes down to temp.
I will then, likely, be sidetracked by something. It's fully carbed so whether it sits in the fridge for 12 hours or 16 hours doesn't matter, it's ready to go!
Also, there are two pseudo-bennies that come along with carbonating before chilling. Not that there is anything wrong with chilling first, it's just that: If you are in a rush and the party that you brewed the beer for is tomorrow, I would rather sample warm carbonated beer now and determine if everything tastes okay before day-of-party. Secondly, and the scenario is similar, if you are rushing to get into bottles so that you can send some cases down to a party in Madison (Go Bucky) that is tomorrow, it can now be done by saving the (typical) 12 hours of chilling time... You know... If you HAVE to... Since bottling room temp beer is a foam nightmare. (But can be done with patience and cold bottles)
Blah blah blah, this is just my ridiculous rationale. Realistically, if you can afford the patience and time, the "best" way to do it would probably be to chill then carbonate and that is really only for one silly reason: The gauges on your regulator are most accurate in the center (noon) position. Assuming that you have 30 psi gauges, you would get your most ACCURATE carbonation for your average beer when carbonating around 40F
Oh, and (I'm really rambling this morning, when is the coffee going to kick in?) one minor correction to your above process. "Rock for 15 minutes" should be a generalization. In reality, you will "rock until you no longer hear the regulator humming, no longer hear bubbling inside the keg, and quite possibly have beer visible inside your CO2 line."
Cheers!