This actually reminds me a bit of talk I've seen in pagan circles over the meaning of sacrifice. The general consensus being that 'sacrifice' has nothing to do with what it is but what you put into it. The amount of work you put into obtaining it or making it and how much you're giving up by giving it to the gods, or whoever it's for, is what counts. I think that holds up in the idea for the hunt quite well in that taking down an animal is no easy task and giving it up rather than keeping it for yourself is, well, a sacrifice.
In this instance it can be applied more in that it's a sacrifice for Diana to kill that deer because she didn't want to harm it. But maybe I'm looking to much into this. Though, since you don't mention it, was there any indication that they ate the deer? *is honestly curious* Because unlike Native Americans, the Greeks/Romans were more likely to set it up and burn it than eat it because there it had nothing to do with the 'circle of life' but actually sacrifing. Just wondering how this was written to play out.