Damage caused by dealership

First, let me note that I am *not* an expert on 4WD, so I will take it on faith that you are correct and that all four tires would need to be replaced. 
That said, *if* they were at fault, then yes, they should have to pay to replace all 4 tires; when someone damages your property and is at fault in doing so, they become liable for the full cost to repair the damage, even if it involves a larger repair/replacement than the actual damage. (To analogize: say that someone negligently damages the hardwood floor in your kitchen: even if they only damaged a 2' x 2' section of planks, they are liable for the cost to not just repair that wood, but to refinish the entire floor so the color matches.)
The issue is, can you prove they were at fault? You would have to prove that the damage occured because they were unreasonably careless or negligent in some way. That may be difficult: for example, say that the damage was due to nail--how can you "prove" it occured at their dealership, because they left nails lying around, and did not occur before or after (e.g. before you dropped the car off, or after you picked it up). You will have your testimony as to the timing of the fault, but if you did not actually see (and ideally photograph) a nail sticking out of your tire at the dealership, or document that gash to them while you were there, it may be difficult to prove in a court (since if they won't pay voluntarily, you'd have to sue) that they caused the damage.