The problem here is that the guy is trying to get around the system because he probably wants to avoid the insurance claim going against him (which happens when there are claims filed). But that's not your problem. If you reported the accident to your insurance company, the legal obligation is theirs to handle the compensation. The other person can file a claim with your insurer, and that's it - they sort it out with them.
The practical trick that nobody mentioned: document everything in writing. If he pressures you for direct payment, send him an email or text clearly stating that you've already reported the accident and that the insurance company is responsible. Keep that. If he insists, you can always show the police that he tried to coerce you into making an out-of-pocket deal (which is legally questionable). Plus, if you agree to pay everything out of pocket now, you can't ask the insurance company for a refund later - it's all or nothing.
The smartest strategy is to contact your insurance company again, tell them the other guy is pressuring you, and ask for direct contact info for the claims handler or a claim number to give him. Let them explain to him how it works. He might not like it, but that's not your problem - you're the one he hit, and you already reported it.