I screwed up and didn't notice that I could direct Rabbit while that notification popped up, so he either died or ran away somehow (I think the former is what happened). How would that situation have been different if I had saved him?
The correct answer is absolutely nothing.
He either tells you where the ambush place is, or you retrieve the very same information automatically from his cellphone, which he drops when killed. Apart from that one dialog when he escapes ("thank you man, here's the ambush site") gameplay is not affected.