I will not pretend that this is not related to the bike battle for the 7 tails, because frankly it is. Can a ninja still use ordinary techniques without a heart...
Well, a Jashinist without a heart would probably lose a bike race, cause you know, would be too busy crying out in pleasure on the bike as a result of all that pain. But, you know, Athos is not a jashinist, just a guy with jashinist immortality, so you'd definitely have to get the judge to rule on whether not having a heart would affect his pedaling and stuff.
I ask because despite having no heart, Athos is claiming to be able to use a space-time barrier and claiming to be on the road to recovery, odd since even Kakuzu was left quite vulnerable without any hearts.
I am sure the rasenshuriken blades cutting through his chakra network had nothing to do that.
Shinobis Chakra nature is tied to their heart as seen by the Earth grudge fear. Bijou cloak I see, but healing vital organs and regular jutsu?
Well, Hidan pierced his own heart and regenerated the damage from that, but got cut up into pieces and did not regenerate arms and legs to dig himself out of the hole Shikamaru made for him. So if anything Jashinist immortality is linked to the head/brain region, not per say the heart, so while it would probably hurt and his blood pressure would drop to 0, over a long period of time he probably could recover.
In the context of this fight, it is unlikely that Athos will "die" conventionally, as even Hidan survived at least to the Great Shinobi War, perhaps longer if the naruto.wikia page is to be believed, and he was left as just a head. Of course if neither surrenders theoretically this fight could go on for a very long time.
While I am not sure the 9-tails seal is in serious danger of breaking at this point in the fight, that is probably the only other way to get Athos much closer to dying at this point, and I am not really sure what if anything would help with that.
The big question is, where is the judge? It does not seem like either of you are going to be calling it quits on your own anytime soon, so if a back and forth waiting game is what the fight is going to turn into, the judge needs to be either switched out or kept on call.