It sounds like the other guy may be trying to commit insurance fraud. If his policy was expired at the time of the accident than he was an uninsured driver at the time of the accident, period. He can't go in and pay after the fact in order to have the accident retroactively covered. The forms people sign to reinstate the policy asks about if you've had any accidents when the policy was lapsed. Also if you have a police report about the accident it will show the date and time which will also show he wasn't insured when the accident occurred. You don't want to get wrapped up in lying to an insurance company about when an accident occurred so that they pay for the damages.
Your best bet is to contact your insurance company and let them know exactly what happened. You'll have to pay any applicable uninsured motorist coverage deductible but it is the right way to go about handling this accident. The deductible might be $100 since you know who hit you. If this had been an incident where you didn't have his information it might have been a $300 deductible. Your insurance company will go after him to get reimbursed for what they pay out. Most insurance companies will pay you back that $100 deductible if and when they collect from him.