Now I think it might be better if the plan were pushed forward a bit, but for a simpler unmanned probe.
This would be a bit simpler, and the probe would arrive right as the comet does.
Now you say this wouldn't require Falcon Heavy, but hear me out.
New Horizons will be able to get to Pluto quickly because the ration of probe to the rocket fuel under it was exaggerated even more--a top end Atlas with a Centaur and a solid upper stage both.
So if we have a fuel fat upper stage and a very small probe, would that be enough to slingshot around Mars' backside and then catch up with the comet, to fly along with it? I still don't think the trajectories would allow it--but that might be something to bring up elsewhere to get the question answered.
Usually, to catch up with something, you have to do a very long matching trajectory as Rosetta is doing, but with Mars there with the comet, could you do a slingshot so as to have a probe ride with that comet--or at least do a flyby?