I voted for cars being invincible tanks, minus some cosmetic damage. In this game you are thrown around a lot and sometimes so hard that taking those impacts into account for damage assessment should probably vaporize you compared to a standard rocket-in-the-back explosion.
In the past i used to develop mods for Unreal Tournament 2004 and i remember there was something called "network replication" (which basically was the synchronization between clients). So i suppose that, since it seems the decision is to make the damage only cosmetic, you could introduce an option to allow people to hide these effects during network replication so that you could see the cars staying unscratched in your client, while others would see cars with damage effects applied to them. But that's just a suggestion, maybe something to be introduced at a later date if you want...
Regarding the damage during an arena combat: i wonder if the damage impacting the performace of your car during an arena fight would push the thing out of balance because people starts to be impaired in their movements toward the pickups. Again, in unreal tournament, you take damage but are in no way impaired because mobility is key to staying alive in that game.
That is, of course, if the idea is to have cars needing to reach pickups for weapons.
Afterall, isn't it easier to not have to deal with the physics of how a car, capable of going from 0 to supersonic in seconds, copes with having a blown up tire?
TL;DR: maybe an option to hide the damage effects client-side (since they don't affect the handling) and i think arenas should retain people's capability to go fetch upgrades without impairment to preserve balance.