Well, let me explain my intuition behind my objection, even if there's a reason why it might be wrong in this case.

I am, in general, skeptical of claims about Pareto-improvements between agents with fundamentally opposed goals (as distinguished from merely different goals, some of which are opposed). Each side has a chance to defect from this agreement to take utility from the other.

It's a quite famliar case for two people to recognize that they can submit their disagreement to an arbitrator who will render a verdict and save them the costs of trying to tip the conflict in their favor. But to the extent that one side believes the verdict will favor the other, then that side will start to increase the conflict-resolution costs if it will get a better result at the cost of the other. For if a result favors one side, then a fundamentally opposed other side should see that it wants less of this.

So any such agreement, like the one between foxes and rabbits, presents an opportunity for one side to abuse the other's concessions to take some of the utility at the cost of total utility. In this case, since the rabbit is getting the benefit of spending the energy of a full chase without spending that energy, the fox has reason to prevent it from being able to make the conversion. The method I originally gave shows one way.

Another way foxes could abuse the strategy is to hunt in packs. Then, when the rabbit spots one of them and plans to run one direction, it will be ill-prepared for if another fox is ready to chase from another direction (optimally, the opposite) -- and gives away its location! (Another fox just has to be ready to spring for any rabbit that stands and looks at something else.)

So even if the "stand and look"/"give up" pattern is observed, I think the situation is more complicated, and there are more factors at play than timtyler listed.