Claim for what?
It depends really.
On the whole, it would not be proper to arrest both parties. That would inevitably mean that the victim of every fight is arrested.
In any event, if they are investigating you then they are not going to take a crime report about him arising from the same incident.
I think overall the reason you were arrested is that he was making an allegation of crime. If all you were alleging is that he was following you then, even if the driving was of a poor standard, that would not be an arrestable offence.
For that reason, I'm not sure what you could claim for? You are not describing a crime or a civil tort so far.
Can I clarify anything for you?
Well, it doesn't matter whether he was or not. In the course of a fight people do tend to become intimidating.
Threat of violence only constitutes an assault by apprehension if there was an immediate apprehension of violence and obviously the police would normally investigate the battery rather than the assault by apprehension. Doing otherwise would be an affront to common sense.
Complainants do usually exaggerate.
How you disprove it is another matter. There are no magic wands. It depends on what evidence you have available. I suspect in a case such as this it will come down to giving evidence in a way that is believed. How that is achieved is really something only you can do.
That is always supposing it leads to a charge.