My fellow volunteers appear to have saved me a lot of work today.
Just a couple of points, though.
You are confusing "no-fault state" with whether somebody is at fault in an accident. "No-fault state" refers to the necessity of injuries being covered by your own insurance regardless of fault.
When it's just damage to auto you have the option of pursuing the claim on your own policy or pursuing the whole cost of repair from the other driver.
Generally, when your own insurance company pays for the damage to your car it goes after the other driver for the full amount of your claim (that's called subrogation) including the deductible and refunds your deductible when the money comes in.
First thing I suggest is that you call your claim rep and find out if a decision has been made to subrogate and whether they are pursuing the full amount including your deductible.
If yes, you need do nothing but wait. You'll be notified if they are successful or not.
It's always possible that the claim rep decides not to subrogate, it's their option, it's not up to you. If the claim rep has chosen not to subrogate (or subrogation fails), you are on your own for your deductible and I agree that small claims court is probably the best way to go after it.
Don't be afraid of threats of a lawsuit. If you got sued over this, you just turn it over to your insurance company and it gets taken care of. Nothing to be concerned with and nothing to stop you from suing on your own.
Now for your questions:
Does anyone have any general tips for working this out?
Small claims court.
Is it worth trying to work with his insurance company?
His insurance company already denied the claim. There's nothing to work out. You'll have to sue the driver.
Does anyone agree with the other insurance company?
Of course not. But the other insurance company is not your insurance company and it owes you nothing until and unless a court of law says so.
Does the fact that no ticket was issued translate into no fault for the accident?
It's the details of the accident itself that counts. And it's true that a driver coming out of a driveway or private road must yield to traffic on the main road. Your photos will probably go a long way in your favor in court.
Is it possible for me to request his statement from his own insurance company for documentation?
Sure, you can request it, but you won't get it until you sue, if you get it all. Small claims court rules generally don't permit "discovery" (the exchange of information and documents prior to trial) so what's likely to happen is you both get to court and tell your stories and the judge makes a decision.
Here's a link to small claims in MI: