One of the side effects of a fake insurance slip (which is equal to riding without insurance) is that since we have a no-fault insurance system, if you are in a collision that is someone else's fault, you have to file the claim with YOUR insurance company, and it's up to them to recover damages (in certain situations) from the other driver's insurance company. If someone else hits you and you don't have an insurance company to file a claim with, you are skreeewwwed AND you will get a driving without insurance ticket ($5000 fine).
In a situation like this where it is likely that the damage will be in the thousands, I wouldn't count on the other driver coughing up the cash. Sure, go ahead and take their information, but also go ahead and file the incident with the collision reporting center and prepare to go through insurance. If they come up with the cash on YOUR terms prior to going through with a claim, fine. If not ... go through insurance.
I had a situation like that with my car a couple years ago. Driver who hit me didn't want to go through insurance ... I took the car to the only place I would trust to repair it (a genuine Mercedes-Benz collision repair shop) ... Driver who hit me wanted me to get three prices and pick the lowest ... sorry, nope, the dealer's repairing it with genuine M-B parts, "unless you pay the shop $X in cash tomorrow, it's going through insurance" ... sure enough, insurance paid.