Since you haven't gotten other responses, I'll try to help:
When we were there last month, the room service menu mentioned nothing about the signature restaurants. My guess is that you may be able to order it to go (since they are capable of packing up leftovers for you when you dine there), but I don't believe room service would bring it to you. Factor in the mandatory service charge AND mandatory gratuity for room service, and I would just call down to Shula's directly and try to order it to go.